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THE

MECHANICAL THEORY

OP

HEAT.





THE

MECHANICAL THEOKYOP

HEAT.

BY

R CLAUSIUS.

TRANSLATED BY

WALTER E. BROWNE, M.A.,LATE FELLOW OF IRINIIT COLLEOS, OAMBKIDGB.

Hontjon

MACMILLAN AND CO.

1879





TRANSLATOE'S PREFACE.

The following translation was undertaken at the instance

of Dr T. Archer Hirst, F.R.S., the translator of the first

collected edition (mentioned in the Author's Preface below)

of Professor Clausius' papers on the Mechanical Theory of

Heat. The former work has however been so completely re-

written by Professor Clausius, that Dr Hirst's translation has

been found scarcely anywhere available ;

and I must there-

fore accept the full responsibility of the present publication.

I trust it may be found to supply a want which I have

reason to believe has been felt, namely, that of a systematic

and connected treatise on Thermodynamics, for use in

Universities and Colleges, and among advanced students

generally. With the view of rendering it more complete for

this purpose, I have added, with the consent of Professor

Clausius, three short appendices on points which he had left

unnoticed, but which still seemed of interest, at any rate

to English readers. These are, (1) The Thermo-elastic pro-



vi translatoe's preface.

perties of Solids; (2) The application of Thermo-dynamical

principles to Capillarity; (3) The Continuity of the Liquid

and Gaseous states of Matter. My best thanks are due to

Dr John Hopkinson, F.E..S., both for the suggestion of these

three points, and also for the original and very elegant

investigation from first principles, contained in the first

Appendix, and in the commencement of the second. My

thanks are also due to Lord Rayleigh, F.R.S., E. J. Routh,

Esq., and Professor James Stuart, for kindly looking through

the first proof of the translation, and for various valuable

suggestions made in connection with it.

WALTER E. BROWNE.

10, ViCTOKiA Chambees, Wesimiksteb,

November, 1879.



AUTHOR'S PREFACE.

Many representations having been made to the author from

different quarters that the numerous papers "On the Me-

chanical Theory of Heat," which he hadpublished

at different

times during a series of years, were inaccessible to many who,

from the widespread interest now felt in this theory, were

anxious to study them, he undertook some years back to

publish a complete collection of his papers relating to the

subject.

As a fresh edition of this book has now become necessary,

he has determined to give it an entirely new form. The

Mechanical Theory of Heat, in its present development, forms

already an extensive and independent branch of science.

But it is not easy 'to study such a subject from a series of

separate papers, which, having been published at different

times, are unconnected in their form, although they agree

in their .contents. Notes and additions, however freely used

to explain and supplement the papers, do not wholly over-

come the difficulty. The author, therefore, thought it best

so to re-model the papers that they might form a connected

whole, and enable the work to become a text-book of the

science. He felt himself the more bound to do this because



Vlll AUTHOES PREFACE.

his long experience as a lecturer on the.Mechanical Theory

of Heat at a Polytechnic School and at several Universities

had taught him how the subject-matter should be arranged

and represented, so as to render the new view and the new

method of calculation adopted . in this somewhat difficult

theory the more readily intelligible. This plan also enabled

him to make use of the investigations of other writers, and

by that means to give the subject greater completeness and

finish. These authorities of course have been in every case

duly recognized by name. During the ten years which haveelapsed since the first volume of papers appeared, many fresh

investigations into the Mechanical Theory of Heat have been

published, and as these have also been discussed, the contents

of the volume have been considerably increased.

Therefore in submitting to the public this, the first part

of his new investigation of the Mechanical Theory of Heat,

the author feels that, although it owes its origin to the second

edition of his former volume, still, as it contains so much

that is fresh, he may in many respects venture to call it a new

work.

R. OLAUSIUS.,

Bonn, December, ]87o.



TABLE OF CONTENTS.

MATHEMATICAL INTRODUCTION.

ON MECHANICAL WOKK, ON ENERGY, AND ON THE TREATMENT OP

NON-INTEGEABLE DIFFERENTIAL EQUATIONS.

§3.

Defiuition and Measurement of Mechanical WorkMathematical Determination of the Work done by variable com-

ponents of ForceIntegration of the Differential Equations for Work done

Geometrical interpretation of the foregoing results, and obaer-

Tations on Partial Differential Coefficients

Extension of the foregoing to three Dimensions

On the Brgal

General Extension of the foregoing

Eelation between Work and Vis Viva .

On Energy

PAGE

1

9

11

13

17

19

CHAPTER I.

FIRST MAIN PRINCIPLE OP THE MECHANICAL THEORY OP HEAT.

EQUIVALENCE OF HEAT AND WORK.

§ 1. Starting Point of the theory 21

§ 2. Positive and Negative Values of Mechanical Work ... 22

§ 3. Expression for the First Main Principle 23

§ 4. Numerical Eelation between Heat and Work .... 24

§ 5. The Mechanical Unit of Heat 25

§ 6. Development of the First Main Principle 27

§ 7. Different Conditions of the quantities J, W, and K . . .27§ 8. Energy of the Body 30

§ 9. Equations for Finite Changes of Condition—Cyclical Processes . 32

§ 10. Total Heat—^Latent and Specific Heat 33

§ 11. Expression for the External Work in a particular case . . 35



X CONTENTS.

CHAPTER II.

ON PERFECT GASES.

PAOE

§ 1. The Gaseous Condition of Bodies 38

§ 2. Approximate Principle as to Heat absorbed by Gases ... 42

§ 3. On the Form which the equation expressing the first Main Prin-

ciple assumes in the case of Perfect Gases .... 43

§ 4. Deduction as to the two Specific Heats and transformation of the

foregoing equations 46

§ 5. Eelation between the two Specific Heats, and its appUcatiou to

calculate the Mechanical Equivalent of Heat .... 48

§ 6. Various Formulse relating to the Specific Heats of Gases . . 52§ 7. Numerical Calculation of the Specific Heat at Constant Volume . 56

§ 8. Integration of the Differential Equations which express the First

Main Principle in the case of Gases 60

§ 9. Determination of the External Work done during the change of

Volume of a Gas 64

CHAPTER III.

SECOND MAIN PEINCIPLE OF THE MECHANICAL THEORY OF H!eAT.

§ 1. Description of a special form of Cyclical Process .... 69

§ 2. Eesult of the CycHoal Process 71

§ 3. Cyclical Process in the case of a body composed partly of Liquid

and partly of Vapour 73

§ 4. Camot's view as to the work performed during the Cyclical

Process 76

§5.

NewFundamental Principle concerning

Heat. . .

'.

78§ 6. Proof that the Eelation between the Heat carried over, and that

converted into work, is independent of the matter which

forms the medium of the change ...... 79

§ 7. Determination of the Function (Tj Tj) 81

§ 8. Cyclical Processes of a more complicated character ... 84

§ 9. Cyclical Processes in which taking in of Heat and change of

Temperature take place simultaneously 87

CHAPTER IV.

THE SECOND MAIN PEINCIPLE UNDER ANOTHER FORM; OR PRIN-

CIPLE OF THE EQUIVALENCE OF TEANSFORJIATIONS.

§ 1. On the different kinds of Transformations 91

§ 2. On a Cyclical Process of Special Form 92

§ 3. On Equivalent Transformations 86



CONTENTS. XI

PAOB

§ 4. Equivalence Values of the Transformations 97

§ 5. Combined value of all the Transformations wliieh take place

in a single CycHeal Process . . . . , .'

. 101

§ 6. Proof that in a reversible Cyclical Process the total value of all

the Transformations must be equal to nothing . . . 102§ 7. On the Temperatures of the various quantities of Heat ; and the

Entropy of the body 106

§ 8. On the Function of Temperature, t . . . . . .107

. CHAPTER V.

FORMATION QF THE TWO FUNDAMENTAL EQUATIOlfS.

§ 1. Discussion of the Variables which determine the Condition of

the body . , 110

§ 2. Elimination of the quantities U and S from the two fundamental

equations . ; . 112

§ 3. Case inwhich the Temperature is taken as one of the Independent

Variables 115

§ 4. Particular Assumptions as to the External Forces . . . 116

§ 5. Frequently occurring forms of the differential equations . . 118

§ 6. Equations in the case of a body which undergoes a partial change

in its condition of aggregation 119

§ 7. Clapeyron's Equation and Carnot's Function .... 121

CHAPTER VI.

APPLICATIOjST of the mechanical theory of heat to SATURATED

VAPOUR.

§ 1. Fundamental equations for Saturated Vapour .... 126

§ 2. Specific Heat of Saturated Steam 129

§ 3. Numerical Value of h for Steam 133

§ i. Numerical Value of h for other vapours 135

§ 5. Specific Heat of Saturated Steam as proved by experiment . . 139

g 6. Specific Volume of Saturated Vapour 142

I 7. Departure &om the law of Mariotte and Gay-Lussac in the case

of Saturated Steam ........ 143

5 8. Differential-coefficients of — 148

§9. A Formula to determine the Specific volume of Saturated Steam,

and its comparison with Experiment 150

110. Determination of the Mechanical Equivalent of Heat from the

behaviour of Saturated Steam 155



Xll CONTENTS.

PAGE

§ 11. Complete Differential Ecâation for Q in the case of a body

composed both of Liquid and Vapour 156

§ 12. Change of the Gaseous portion of the mass .... 157

§ 13. Belation between Volume and Temperature .... 159

§ 14. Determination of the Work as a function of Temperature . . 160

CHAPTER VII.

FUSION AND VAPORIZATION OF SOLID BODIES.

i1. Fundamental Equations for the process of Fusion . . . 163*

i 2. Eelation between Pressure and Temperature of Fusion. . 167

i3. Experimental Verification of the Foregoing Eesult . . . 168

i4. Experiments on Substances which expand during Fusion . . 169

i5. Eelation between the Heat contained in Fusion and the Tempe-

rature of Fusion 171

I6. Passage from the solid to the gaseous condition .... 172

CHAPTEE VIII.

ON HOMOGENEOUS BODIES.

§ 1. Changes of Condition without Change in the Condition of Aggre-

gation, 175

§ 2. Improved Denotation for the differential coefficients . . . 176

§ 8. Eelations between the Differential Coefficients of Pressxire,

Volume, and Temperature 177

§ 4. Complete differential equations for Q 178§ 5. Specifie Heat at constant volume and constant pressure . . 180

§ 6. Specific Heats under other oiroumstanees 184

§ 7. laentropic Variations of a body 187

§ 8. Special form of the fundamental equations for an Expanded Eod 188

§ 9. Alteration of temperature during the extension of the rod . . 190

§ 10. Further deductions from the equations 192

CHAPTEE IX.

DETEKMINATION OF ENERGY AND ENTROPY.

i

1. General equations 195

I2. Differential equations for the case in which only reversible

changes take place, and in which the condition of the Body is

determined by two Independent Variables .... 197



CONTENTS. Xlll

PAGE

§ 3. Introdnction of the Temperature as one of the Independent

YatiaWes 200

§ 4. Special case of the differential equations on the assumption that

the only External Force is a uniform surface pressure . . 203

§ 5. Application of the foregoing equations to Homogeneous Bodiesand in particular to perfect Gases 20S

§ 6. Application of the equations to a body composed of matter in

two States of Aggregation 207

§ 7. Kelation of the expressions Dxy and Axy 209

CHAPTER X.

ON NON-BBVEESIBLE PKOCESSES.

§ 1. Completion of the Mathematical Expression for the Second

Main Principle 212

§ 2. Magnitude of the Uncompensated Transformation . . . 214

§ 3. Expansion of a Gas unaccompanied by Work .... 215

§ 4. Expansion of a Gas doing partial work 218

§5. Methods of experiment used by Thomson and Joule . . . 220

§ 6. Deyelopment of the equations relating to the above method . 221

§ 7. Besults of the experiments, and the equations of Elasticity for

the gases, as deduced therefrom 224

§ 8. On the Behaviour of Vapour during expansion and under various

circumstances 228

CHAPTER XL

APPLICATION OF THE THEOKT OF HEAT TO THE STEAM-ENGINE.

i1. Necessity of a new investigation into the theory of the Steam-

Bngiue 234

j2. On the Action of the Steam-Engine 236

I3. Assumptions for the purpose of SimpMoation .... 238

I

4. Determination of the Work done during a single stroke . . 239

i5. Special forms of the expression fovmd in the last section . . 241

\ 6. Imperfections in the construction of the Steam-Engine . . 241

I7. Pambour's Eormulse for the relation between Volume and Pres-

sure 242

J8. Pambour's Determination of the Work done during a single

stroke 244

) 9. Pambour's Value for the Work done per unit-weight of steam . 247

i 10. Changes in the Steam during its passage from the Boiler into

the Cylinder 248



XIV CONTENTS,

PAOE

§ 11. Divergence of the Eesnlts obtained from Pambour's Assumption 251

§ 12. Determination of the work done during one stroke, taking iato

consideration the imperfections abeady noticed . . » 252

§ 13. Pressure of Steam in the Cylinder during the different Stages ,

of the Process and corresponding Simplifications of the

Equations 255

§ 14. Substitution of the Vohune for the corresponding Tempera-

ture in certain cases . . 257

§ 15. Work per Unit-weight of Steam 259

§ 16. Treatment of the Equations 259

§ 17. Determination of -^ and of the product Tg .... 260

§ 18. Introduction of other measures of Pressure and Heat . . 261

§ 19. Determination of the temperatures T^ and Ta . . . . 262

§ 20. Determination of c and r . . . ' 264

§ 21. Special Form of Equation (32) for an Engine working without

expansion 265

§ 22. Numerical values of the constants 267

§ 23. The least possible value of V and the corresponding amount of

Work 268

§ 24. Calculation of the work for other values of F . . . . 269

§ 25. Work done for a given value of V by an engine with expansion . 271

§ 26. Summary of various cases relating to the working of the engine 273

§ 27. Work done per unit of heat delivered from the Source of Heat . 274

§ 28. Friction . 275

§ 29. General investigation of the action of Thermo-Dynamio Engines

and of its relation to a Cyclical Process 276

§ 30. Equations for the work done during any cychoal process . . 279

§ 31. AppHoation of the above equations to the limiting case in which

the Cyclical Process in a Steam-Engine is reversible . . 281

§ 32. Another form of the last expression 282

§ 33. Influence of the temperature of the Source of Heat . . . 284

§ 34. Example of the application of the Method of Subtraction . . 286

CHAPTEE XII.

ON THE CONCENTRATION OP EATS OF LIGHT AND HEAT, AND ON

THE LIMITS OF ITS ACTION.

§1. Object of the

investigation 293I. Beasons why the ordinary method of determining the mutual

radiation of two surfaces does not extend to the present

case 296

g 2. Limitation of the treatment to perfectly black bodies and to

homogeneous and unpolarized rays of heat .... 296

§ 3. Kirchhoff's Formula for the mutual radiation of two Elements of

surface 297

§ 4. Indetermiuateness of the Formula in the case of the Concen-

tration of rays 300



CONTENTS. XV

PAGE

II. Determination of corresponding points and corresponding

Elements of Surface in three planes cut by the rays . 301

§ 5. Equations between the co-ordinates of the points in which a ray

cuts three given planes 301

§6. The relation of corresponding

elements of surface. . . 304

§ 7. Various fractions formed out of six quantities to express the

Eolations between Corresponding Elements .... .SC9

III. Determination of the mutual radiation when there is no

concentration of rays ....... 310

§ 8. Magnitude of the element of surface corresponding to ds^ on a

plane in a particular position....... 310

§ 9. Expressions for the quantities of heat which da, and ds, radiate

to each other 812

§ 10. Eadiation as dependent on the surrounding medium . . . 314

IV. Determination of the mutual radiation of two Elements of

Surface In the case when one is the Optical Image of the

other 316

§ 11. Relations between B,D,F,a.ndiE 316

§ 12. Application of A and C to determine the relation between the

elements of surface 318

§ 18. Eelation between the quantities of Heat which ds, and ds^

radiate to each other 319

V. Eelation between the Increment of Area and the Eatio of

the two solid angles of an Elementaiy Pencil of Bays . 321

§ 14. Statement of the proportions for this case 321

VI. General determination of the mutual radiation of two sur-

faces in the case where any concentration whatever maytake place 324

§ 15. General view of the concentration of rays 324

§ 16. Mutual radiation of an element of surface and of a finite surface

through an element of an intermediate plane . . . . 3r 5

§ 17. Mutual radiation of entire surfaces 3'. 8

§ 18. Consideration of various collateral circumstances . . . 329

§ 19. Summary of results 330

CHAPTER XIII.

DISCUSSIONS ON THE MECHANICAL THEORY OF HEAT AS HEKE

DEVELOPED AND ON ITS FOUNDATIONS.

j 1. Different views of the relation between Heat and Work . . 332

i2. Papers on the subject by Thomson and the author . . . 3S3

i

3. On Eankine's Paper and Thomson's Second Paper . . . 3i?4

i4. Holtzmaan's Objections 337

i5. Decher's Objections 339

i

6. Fundamental principle on which the author's proof of the second

main principle rests . 340

i7, Zeuner's first treatment of the subject 341

i8. Zeuner's second treatment of the subject 342



XVI CONTENTS.

PAGE

§ 9. Eankine's treatment of the subject 345

§ 10. Him's Objections ... 348

§ 11. Wand's Objections . . ... 353

§ 12. Tait'a Objeetions 860

Appenhix I. Thermo-elastie Properties of Solids .... 363

II. CapiUarity 369

„ III. Continuity of the Liquid and Gaseous states . . . 373



ON THE MECHANICAL THEORY OP HEAT.

MATHEMATICAL INTRODUCTION.

ON MECHANICAL WORK, ON ENEHGT, AND ON THETREATMENT OF NON-INTEGRABLE DIFFERENTIAL

EQUATIONS.

§ 1. JOefinition and Measurement of Mechanical Work.

Every force tends to give fiiotion to the body on which

it acts; but it may be prevented from doing so by other

opposing forces, so that equilibrium results, and the bodyreniains at rest. In this case "the force performs no work.

But as soon as the body moves under the influence of the

force, Work is performed.

In order to investigate the subject of Work under thesimplest possible conditions, we may assume that instead

of an extended body the force acts upon a single material

point. If this point, which we may call p, travels in the

same straight line in which the force tends to move it,

then the product of the force and the distance moved

through is the mechanical work which the force performs

during the motion." If on the other hand the motion of

the:point is in any other direction than the line of action

of the force, then ibhe work performed is represented by

the product of the distance moved through, and the com-

ponent of the force resolved in the direction of motion,

This component of force in the line of motion may be

piopitive or negative in sign, according as it tends in th§

c. 1 '





MATHEMATICAL INTRODUCTION. 3

the one hand the force need not itself he the same at diffe-

rent points of space ; and on the other, although the force

may remain constant throughout, yet, if the path be not

straight but curved, the component of force in the direction

of motion will still vary. For this reason it is allowable to ex-

press work done by a simple product, only when the distance

traversed is indefinitely small, i.e. for an element of space.

Let ds be an element of space, and S the component in

the direction of ds of the force acting on the point p. Wehave then the following equation to obtain dW, the work

done during the movement through the indefinitely small

space dsdW=Sds (1).

If P be the total resultant force acting on the point p, and

^ the angle which the direction of this resultant makes with

the direction of motion at the point under consideration,

then

S =P cos<f),

whence we have, by (1),

dW =P cos ^ds (2).

It is convenient for calculation to employ a system of

rectangular co-ordinates, and to consider the projections of

the element of space upon the axes of co-ordinates, and the

components of force as resolved parallel to those axes.

For the sake of simplicity we will assume that the motion

takes place in a plane in which both the initial direction of

motion and the line of force are situated. We will employ

rectangular axes of co-ordinates lying in this plane, and will

call X and y the co-ordinates of the moving point ^ at a

given time. If the point moves from this position in the

plane of co-ordinates through an indefinitely small space ds,

the projections of this motion on the axes will be called cfe

and dy, and will be positive or negative, according as theco-ordinates x and y are increased or diminished by the

motion. The components of the force F, resolved in the

directions of the axes, will be called X and Y. Then, if a

and b are the cosines of the angles which the line of force

makes with the axes of a; and y respectively, we have

x'=aP; Y=bP.

1—2



4 ON THE MECHANICAL THEOET OF HEAT.

Again, if a and y3 are the cosines of the angles wliich the

element of space ds makes with the axes, we have

dx = ads ; dy = ^ds.

¥rom these equations we obtain

Xdx + Ydy = {aa. + &/3) Pds.

JBut by Analytical Geometry we know that

aa + i^ = cos ^,

where ^ is the angle between the line of force, and theelement .of space : hence

Xdx + Ydy = cos ^ Pds,

.and therefore by equation (2),

dW=Xdx+ Ydy (3).

This being the equation for the work done during an indefi-

nitely small motion, we must integrate it to determine the

work done during a motion of finite extent.

§ 3. Integration of the Differential Equation for Work-

done.

'

In the integration of a differential equation of the form

given in equation (3), in which X and Y are functions of xand y, and which may therefore be written in the form

dW=<f>(!^)dx+ y{r(a^\dy (3a),

a distinction has to be drawn, which is of great importance,

not only for this particular case, but also for the equations

which occur later on in the Mechanical Theory of Heat j and

whichwill

therefore be examined here at .some length, sothat in future it will be sufficient simply to refer back to thepresent passage. ;

According to the nature of the functions ^ (my) and\fr (o^), differential equations of the form (3) fall-iSto twoclasses, which differ widely both as to the treatment whichthey require, and the results to which they lead. To the



ilATHEMATICAI/ INTRODUCllOU. 5

first class belong the cases, in which, the fiinctions X and Yfulfil the following condition

dy dx '"" ""^ ''

The second class comprises all cases, in which this condition

is not fulfilled.

If the condition (4) is fulfilled, the expression on the

right-hand side of equation (3) or (3a) becomes immediately

integrable ; for it is the complete differential of some func-

tion of X and y, in which these may be treated as indepen-

dent variables, and which is formed from the equations

dx ,' dy

Thus we obtain at once an equation of the form

w =F {xy) + const (5).

If condition (4) is not fulfilled, the right-hand side of the

equation is not integrable ; and it follows that W cannot be

expressed as a function of x and y, considered as independent

variables. For, if we could put W= Fiacy), we should have

^^dW^ dF{xy)

dib dx '

Y^dW^ dFjxy)

dy dy '

whence it follows that

dX ^ d'Fjxy)

dy dxdy '

dY^ d^Fjxy) ,

dx dydx

But since with a function of two independent variables the

oxdisr of differentiation is immaterial, we may put

dxdy dydx'



6 ON THE MECHANICAL THEORY OF HEAT.

whence it follows that-y- =-5— j «'e- condition (4) is fulfilled

for the functions X and Y; which is contrary to the assump-

tion.

In this case then the integration is impossible, so long as

x and y are considered as independent variables. If however

we assume any fixed relation to hold between these two

quantities, so that one may be expressed as a function of the

other, the integration again becomes possible. For if we

put

/(^2/) = (6),

in which f expresses any function whatever, then by means

of this equation we can eliminate one of the variables and

its differential from the differential equation. (The general

form in which equation (6) is given of course comprises the

special case in which one of the variables is taken as

constant ; its differential then becomes zero, and the variable

itSBlf only appears as part of the constant coefficient). Sup-

posing y to be the variable eliminated, the equation (.3)

takes the form dW=<^{x) dx, which is a simple differential

equation, and gives on integration an equation of the form

w = F {x) \- comi (7).

The two equations (6) and (7) may thus be treated as form-

ing together a solution of the differential equation. As the

form of the function fixy) may be anything whatever, it is

clear that the number of solutions thus to be obtained is

infinite.

The form of equation (7) may of course be modified.

Thus if we had expressed x in terms of y by means of equa-

tion (6); and then eliminated x and dx from the differential

equation, this latter would then have taken the form

dW= <^,{y)dy,

and on integrating we should have had an equation

''^=^1(2')+ const (7a3.

This same equation can be obtained from equation (7) by

substituting y for x in that equation by means of equation (6).

Or, instead of completely eliminating x from (7), we may



MATHEMATICAL ' INTRODUCTION. 7

prefer a partial elimination. For if the function F (cc) con-

tains X several times over in different teriiis, (and if this does

not hold in the original form of the equation, it can be easily

introduced into it by writing instead of x an expression such

as (l — a)x + ax, —^, &c.) then it is possible to substitute

y for X in some of these expressions, and to let x remain in

others. The equation then takes the form

W=^FÎlx,y) + const (76),

which is a more general form, embracing the other two as

special cases. It is of course understood that the three equa^

tions (7), (7a), (76), each of which has no meaning except

when combined with equation (6), are not different solutions,

but different expressions for one and the same solution of

the differential equation.

Instead of equation (6), we may also employ, to integrate

the differential equation (3), another equation of less simple

form, which in addition to the two variables x and y also,

contains W, and which may itself be a differential equation

the simpler form however suffices for our present purpose,

and with this restriction we may sum up the results of this

section as follows.

When the condition of immediate integrability, expressed

by equation (4), is fulfilled, then we can obtain directly an

integral equation of the form

W=F{x,y) + const (A).

When this condition is not fulfilled, we must first assume

some relation between the variables, in order to make inte-

âtion possible; and we shall thereby obtain a system of

two equations of the following form

/(*,2/) = 0, 1 ,gv

W=F{a;.,y) + const.\

^^^'

in which the form of the function F depends not only on

that of the original differential equation, but also on that of

the function/, which may be assumed at pleasure.




§ i. Geometrical interpretation of the foregoing results,

and observations on partial differential coefficients.

The important difference between the results in the two

cases mentioned above is rendered more clear by treating

them geometrically. In so doing we shall for the sake of

simplicity assume that the function F {x, y) in equation (A)

is such that it has only a single value for any one point in

the plane of co-ordinates. We shall also assume that in the

movement of the point p its original and final positions are

known, and given by the co-ordinates x^, y^, and a;,, y^ respec-

tively. Then in the first case we can find an expression for

the work done by the effective force during the motion,:

without needing to know the actual path traversed. For it

is clear, that this work will be expressed, according to con-

dition {A)i by the difference F(x^p^ — F(x„y„). Thus, while

the moving point may pass from one position to the other

by very different paths, the amount of work done by the.force

is wholly independent of these, and is completely known aS

soon as the original and final positions are given.In the second case it is otherwise. In the system of

equations (B), which ' belongs to this case, the first equation

must be treated as the equation to a curve ; and (since the

form of the second depends upon it) the relation between

them may be geometrically expressed by saying that the

work done by the effective force during the motion of the

point p can only be determinedj when the whole of the

curve, on which the point moves, is known. If the original

and final positions are given, the first equation must indeed •

be so chosen, that the curve which corresponds to it may pass

through those two points ; but the number of such possible

curves is infinite, and accordingly, in spite of their coinci-

dence at theii: extremities, they will give an infinite numberof possible quantities of"work done during the motion.

If we assume that the point p describes a closed curve, sothat the final and initial positions coincide, and thus the co-

ordinates x^, y^ have the same value as x„, y^, then in the

first case the total work done is equal to zero: in the second

case, on the Other hand, it need not equal zero, but may haveany value positive or negative.

The case here examined also illustrates the fact that a



.(8).

MATHEMATICAL INTBOBUCTION. 9^.

quantity, which cannot be expressed as a function of x and y(so long as these are taken as independent variables),, mayyet have partial differential coefficients according to x and y,

which are expressed by determinate- functions of those vari-

ables. For it is manifest that, in the strict sense of the

words, the components X and Y must be termed the partial

differential coefficients of the work W according to x and y :

smce, when x increases by dx, y remaining constant, the,

work increases by Xdx ; and when y increases by dy, x re-

maining constant, the work increases by Ydy. Now whether

TF be a quantity generally expressible as a function of x and

.

y,or

one which can only be determined on knowing the pathdescribed by the inoving point, we may always employ the

ordinary notation for the partial differential coefficients of W,and write

Using this notation we may also write the condition (4), the

fulfilment or non-fulfilment of which causes the distinction

between the two modes of treating the differential equation,

in the following form:

d fdW\ _ d rdW\ ,g.

dy\dx) dx\dy )

Thus we may say that the distinction which has to be'

drawn in reference to the quantity W depends on whether

the difference t-(-t- I—

-r- (—7— I is dqual to zpro> or hasdy\dx J dx\dy J

a finite value. , -

§ 5. Extension of the above to three dimensions.

If the point p be not restricted in its movement to one

plane, but left free in space, we then obtain for the element

of work an expression very similar to that given in equation

(3). Let a, b, c be the cosines of the angles which the direc-

tion of the force P> acting on ..the, point, makes with three




rectangular axes of co-ordinates ; then the three components

X, Y, Z of this force will be given by the equations

X=aF, Y=hP, Z=.cP.

Again, let a, A 7 be the cosines of the angles, which the

element of space ds makes with the axes ; then the three,

projections dx, dy, dz of this element on those axes are given

by the equations

dx= ads, dy = ^ds, ds = 'yds.

Hence we have

Xdx + Ydy + Zdz = {pa. + &/3 + cy)Fds.

But if ^ be the angle between the direction ofP and ds^

then

aai + &;8 + cy = cos ^:

hence Xdx + Ydry + Zdz = cos(j>x Pds.

Comparing this with equation (2), we obtain

dW = Xdx+Ydy + Zdz (10).

This is the differential equation for determining the work

done. The quantities X, Y, Z may be any functions what-

ever of the co-ordinates oc,y,z; since whatever may be the

values of these three components at different points in space,

a resultant force P may always be derived from them.

In treating this equation, we must first consider the fol-

lowing three conditions

dX^dY dY^dZ dZ^dXdy dx ^ dz dy ' dx dz ^ '^'

and must enquire whether or not the functions X, Y, Zsatisfy them.

If these three conditions are satisfied, then the expression

on the right-hand side of (11) is the complete differential of

a function of x, y, z, in which these may all be treated as

independent variables. The integration may therefore be at

once effected, and we obtain an equation of the form

TF= P(a!^s) + const (12).



MATHEMATICAL INTEOBUCTION. 11

If we now conceive the point j> to move from a given

initial position (x^, y^, z^ to a given final position (ajj, y^, z^

the work done by the force during the motion will be repre-

sented by

If then we suppose F{x, y, z) to be such that it has only a

single value for any one point in space, the work will be

completely determined by the original and final positions;

and it follows that the work done by the force is always the

same, whatever path may have been followed by the point

in passing from one position to the other.If the three conditions (1) are not satisfied, the integra-

tion cannot be effected in the same general manner. If,

however, the path be known in which the motion takes place,

the integration becomes thereby possible. If in this case

two points are given as the original and final positions, and

various curves are conceived as drawn between these points,

along any of which the point p may move, then for each of

these paths we may obtain a determinate value for the work

dpne; but the values corresponding to these different paths

need not be equal, as in the first case, but on the contrary

are in general, different.

§ 6. OniheErgal.

Inthose cases in which equation

(12)holds, or the work

done can be simply expressed as a function of the co-ordirtates,

this function plays a very important part in our calculations.

Hamilton gave to it the special name of "force function"; a

name applicable also to the more general case where, instead

of a single moving point, any number of such points are

considered, and where the condition is fulfilled that the work

done depends only on the position of the points. In the

later and more extended investigations with regard to the

quantities which are expressed by this function, it has become

needful to introduce a special name for the negative value of

the function, or in other words for that quantity, the svh-

traction of which gives the work performed; and Rankine

proposed for this the term 'potential energy.' This name

sets forth very clearly the character of the quantity ; but it





ItitHEMATlOAt INTRODUCTION,' 1'3

expression in brackets forms a perJFect differential, and wemay write

Xdx+Ydy+ Zdz = -d<f>{p) (16).

The element of work is thus given by the negative differen-

tial of ^ (p) ; whence it follows that 6 (p) is in this case the£rgal.

Again, instead of a single fixed point, we may have anynumber of fixed points tTj, tt^, w„ &c., the distances of whichfrom p^ are p^, p^, p^. Sec, and which exert on it forces

^'(P»)> ^'W. 0'(Pj). &c. Then if, as in equation (14), we as-

sume^j(jo),<^^{p), 4)^(p), &c, to be the integrals of the above

Junctions, we obtain, exactly as in equation (15),

x^. <^'î(pi) ^M #.,(p»)

dx doa dsB

^—^S^W (17).

Similarly r=-|2^(p), Z=-^^t^{p) (I7a),

whence Xdx + Ydy + Zdz = - d% <^ {p) (18).

Thus the sum S ^{p) is here the Ergal.

§ 7. General Extension of theforegoing.

Hitherto we have only considered a single moving point

we will now extend the method to comprise a system com-

posed of any number of moving points, which are in part

acted on by external forces, and in part act mutually on eachother.

If this whole system makes an indefinitely small move-

ment, the forces acting on any One point, which forces we

may conceive as combined into a single resultant, will per-

form a quantity of work which may be represented by the

expression {Xdx + Ydy + Zdz), Hence the sum of all the




work done by all the forces acting in the system may be

represented by an expression of the form

t{Xdx-^ Ydy + Zdz),

in which the summation extends to all the moving points.

This complex expression, like the simpler one treated

above, may have under certain circumstances the important

peculiarity that it is the complete differential of some func-

tion of the co-ordinates of all the moving points ; in which

case we call this function, taken negatively, the Ergal of the

whole system. It follows from this that in a finite move-ment of the system the total work done is simply equal to

the difference between the initial and final values of the

Ergal; and therefore (assuming that the function which,

represents the Ergal is such as to have only one value for one

position of the points) the work done is completely deter-

mined by the initial and final positions of the points, without

its being needful to know the paths, by which these have

moved from one position to the other.

This state of things, which, it is obvious, simplifies greatly

the determination of the work done, occurs when all the

forces acting in the system are central forces, which either

act upon the moving points from fixed points, or are actions

between the moving points themselves.

First, as regards central forces acting from fixed points,

we have already discussed their effect for a single movingpoint ; and this discussion will extend also to the motion of

the whole system of points, since the quantity of work done

in the motion of a number of points is simply equal to the

sum of the quantities of work done in the motion of each

several point. We can therefore express the part of the

Ergal relating to the action of the fixed points, as before, by2 (^ (p), if we only give such an extension to the summation,

that it shall comprise not only as many terms as there are

fixed points, but as many terms as there are combinations of

one fixed and one moving point.

Next as regards the forces acting between the movingpoints themselves, we will for the present consider only twopoints p and p\ w^Hh co-ordinates co, y, z, and x, y', z\



MATHEMATICAL INTKODXJCTION, 15

respectively. If r be tlie distance between these points, wehave

r = J{x'-a=y + (y'-yr+[,'-zy (19).

We may denote the force which the points exert on eachother by/'(r), a positive value being used for attraction, and

a negative for repulsion.

Then the components of the force which the point pexerts in this mutual action are

/W^, /'«^, fir)

and the components of the opposite force exerted by p' are

/'W^', /'W^'. /'(r)i^'.

But by (19), differentiating

dr _ x' — X^dr _ x — x^

dxr ' dx' r '

so that the components of force in the direction of x mayalso be written

and if /(r) be a function such that

f(r)=jf'{r)dr (20),

the foregoing may also be written

-dfjr) , -df(r)^

dx ' dd

Similarly the components in the direction of y may be

written-t?f(r)

.:^/(r).

dy ' dy''

and those in the direction of z

-df{r)^ -df(r)




That part of the total work done in the indefinitely small

motion of the two points, which is due to the two opposite

foTcea arising from their mutual ^,ction, may therefore be

expressed as follows

dz J

But as r depends only on the six quantities x^ ,y, z, oc', y, z,

and f{r) can therefore be a function of these six quantities

only, the expression in brackets is a perfect differential, and

the work done, as far as concerns the mutual action between

the two points, may be simply expressed by tl)e function

In the same way may be expressed the work due to the"

mutual action of every other pair of points ; and the total

work done byall

theforces which the

points exert amongthemselves is expressed by the algebraical suni

-rf/(r)-J/(/)-d/(0-...;

or as it may be otherwise written,

-<^[/W+/(^')+/(0 + -] or -dS/W;

in which the summation must comprise as many terms as

there are combinations of moving points, two and two. Thissum S/(r) is then the part of the Ergal relating to the

mutual and opposite actions of all the moving points.

If we now finally add the two kinds of forces together,

we obtain, for the .to.tal work done in the indefinitely small

motion of the system of points, the equation

2 {Xdx + Ydy + Zdz) =-dt(]>(p)^ dtfir)

= ^d[tcl>{p)+-ZAr)] (21),

whence it foUows that the quantity%<f>{p) + '^f{r) is the

Ergal of the whole of the forces.acting together in ^he systeni,.

The assumption lying at the root of the foregoing analy-

sis, viz. that central forces are the only ones acting, is of

course only one among all .the assumptions mathematically



MATHEMATICAL INTEODUCTION. 17

possible as to the forces; but it forms a case of peculiar

importance, inasmuch as all the forces which occur in nature

may apparently be classed as central forces.

§ 8. Relation between Work and Vis Viva.

Hitherto we have only considered the forces which act on

the points, and the change in position of the points them-

selves ; their masses and their velocities have been left out

of account. We wiU now take these also into consideration.

The equations of motion for a freely moving point of

mass

mare well known to be as follows

If we multiply these equations respectively by

dx 7, dy ,^ dz , •

dt^'' dt^'' It^''

and then add, we obtain

The left-hand side of this equatioii may be transformed

into

m d2dtm<%'<%' dt.

or, if V be the velocity of the point,

î(p.dt=%ldt=d(^v^2 dt dt V2 y

and the equation becomes

<'(f«')=(^S+4!+4)*'^')-

If, instead of a single freely moving point, a whole system

of freely moving points is considered, we shall have for every

n 2




point a similar equation to, the above; and by summation

we shall obtain the following:-

<ffi5^.2(x|+7|+43i. <^«-

Now the quantity 2 n-t)' is* the vis viva of the whole system

of points. If we take a simple expression for the vis viva,

and put

T=-Z''^v' (26),

then the equation becomes

..r=2(z| + r|+zg)c^* (27).

But the right-hand side of this equation is the expression

for the work done during the time dt. Integrate the equa-

tion from an initial time i„ to a time t, and call T^ the vis

viva at time t^: then the resulting equation is

''-^•-/>(^S+4!+^S)* (^«''

the meaning of which may be expressed as follows:

The Wo^h done during any time hy the forces acting upon

a system is equal to the increase of the Yis Viva of the system

during the same time.

In this expression a diminution of Vis Viva is of course

treated as a negative increase.

It was assumed at the commencement that all the points

were moving freely. It may, however, happen that the poihts

are subjected to certain constraints in reference to their

motion. They may be so connected with each other that

the motion of one point shall in part determine the .motionof others ; or there may be external constraints, as for in-

stance, if one of the points is compelled to move in a given

fixed plane, or on a given fixed curve, whence it will natur-

* Translator's Note. The vis viva of a particle is here defined as half

the mass multiplied by the square of the velocity, and not the whole mass,as was formerly the custom.



MATHEMATICAL INTEODUCTION. 19

ally follow that all those points, which are in any connectionwith it, will also be to some extent constrained in their

motion.

If these conditions of constraint can be expressed byequations which contain only the co-ordinates of the points,

it may be proved, by methods which we will not here con-

sider more closely, that the reactions, which are implicitly

comprised in these conditions, perform no work whateverduring the motion of the points ; and therefore the principle

given above, as expressing the relation between Vis Viva andWork done, is true for constrained, as well as for free motion.

It is called the Principle of the Equivalence of Work andVis Viva.

I'M-

§ 9. On Energy.

In equation (28), the work done in the time from t^ to t

is expressed by

in which t is considered as the only independent variable,

and the co-ordinates of the points and the components of the

forces are taken as functions of time only. If these functions

are known (for which it is-requisite that we should know the

whole course of the motion of all the points), then the inte-

gration is always possible, and the work done can also be

determined as a function of the time.

Cases however occur, as we have seen above, in which it

is not necessary to express all the quantities as functions of

one variable, but where the integration may still be effected,

by writing the differential in the form S (Xdx + l^y -t- Zdz),

and considering the co-ordinates therein as independent vari-

ables. For this it is necessary that this expression should

be a perfect differential of some function of the co-ordinates,

or in other words the forces acting on the system must have

an Ergal. This Efgal, which is the negative value of the

above, function, we will denote by a single letter. The letter

U is generally chosen for this purpose in works on Me-

chanics: but in the Mechanical Theory of Heat that letter

is needed to express another quantity, which will enter as

2—2





(21)

CHAPTER I.

FIEST MAIX PRINCIPLE OF THE MECHANICAL THEORY OFHEAT, OR PRINCIPLE OF THE EQUIVALENCE OF HEAT

AND WORK.

§ 1, Nature of Heat.

Until recently it M'as the generally accepted view that

Heat was a special substance, which was present in bodies in

greater or less quantity, and which produced thereby their

higher or lower temperature; which was also sent forth

from bodies, and in that case passed with immense speed

through empty space and through such cavities as ponder-

able bodies contain, in the form of what is called radiant

heat. In later days has arisen the other view that Heat is

in reality a mode of motion. According to this view, the

heat found in bodies and determining their temperature is

treated as being a motion of their ponderable atoms, in

which motion the ether existing within the bodies may also

participate ; and radiant heat is looked upon as an undulatory

motion propagated in that ether.

It is not proposed here to set forth the facts, experiments,

and inferences, through which men have been brought to

this altered view on the subject ; this would entail a refer-

ence here to much which may be better described in its own

place during the course of the book. The conformity with

experience of the results deduced from this new theory will

probably serve better than anything else to establish the

foundations of the theory itself

We will therefore start with the assumption that Heat

consists in a motion of the ultimate particles of bodies and

of ether, and that the quantity of heat is a measure of the

Vis Viva of this motion. The nature of this motion we



22 ox t:he mechanical theoey of heat.

shall not attempt to determine, but shall merely apply to

Heat the principle of the equivalence of Vis Viva and Work,

which applies to motion of every kind ; and thus establish

a principle which may be called the first main Principle of

the Mechanical Theory of Heat.

§ 2. Positive and negative values of Mechanical Work.

In § 1 of the Introduction the meOhanical work done in

the movement of a point under the action of. a force was

defined to be The product of the distance moved through and

of the component of the force resolved in the direction of

motion. The workis

thuspositive if the component of

force in the line of motion lies on the same side of the

initial point as the element of motion, and negative if it falls

on the opposite side. From this definition of the positive

sign of mechanical work follows the principle of the equiva-

lence of Vis Viva and Work, viz. The increase in the Vis

Viya is equal to the work done, or equal to the increase in

total work.

The question may also be looked at from another point

of view. If a material point has once been set in motion, it

can continue this movement, on account of its momentum,

even if the force acting on it tends in a direction opposite to

that of the motion; though its velocity, and therewith its

Vis Viva, will of course be diminishing all- the time. Amaterial point acted on by gravity for example, if it has

received an upward impulse, can continue to move against

the force of gravity, although the latter is continually

diminishing the velocity given by the impulse. In such a

case the work, if considered as work done by the force, is

negative. Conversely however , we may reckon work as

positive in cases where a force is overcome by the momen*tum of a previously acquired motion, as negative in cases

where the point follows the direction of the force. Applying

the form of expression introduced in § 1 of the Introduction,in which the distinction between the two opposite directions

of the component of force is indicated by different words, wemay express the foregoing more simply as follows : we maydetermine that not the work done, but the work destroyed,

by a force shall be reckoned as positive.

On this method of denoting work done, the principle of



EQUIVALENCE OF HEAT AND WORK. 23

the equivalence of Vis Viva and Work takes the following

form : The decrease in ike Vis Viva is equal to the increase in

the Work done, or The sum of the Vis Viva and Work done is

constant. This latter form will be found. very convenient in

what follows.

In the case of such forces as have an Ergal, the meaning

of that quantity was defined (in § 6 of the Introdiôiiqg) in

such a manner that we must say, ' The Work dêîs 'equal

to the decrease in the Ergal.' If we use the method of denot-

ing work JMt ^described, we must say on the contrary, 'The

work dop?îs equal to the increase in the Ergal;'- and if the

constant occurring as one term of the Ergal be determined

in a particular way, we may then regard the Ergal as simply

an expression for.the work done.

§ 3. Expressionfor the first Fundamental Principle.

Having fixed as above what is to be the positive sign for

work done, we may now state as follows the first main

Principle of the Mechanical Theory of Heat.

In all cases where work is produced by heat, a quantity ofheat is consumed proportional to the work done; and inversely,

by the expenditure of the same amount of work tfte same

quantity of heat m,ay be produced.

This follows, on the mechanical conception of heat, from

the equivalence of Vis Viva and Work, and is named The

Principle of the Equivalence of Heat and Work.

If heat is consumed, and work thereby produced, we may

say that heat has transformed itself into work ; and con-

versely, if work is expended and heat thereby produced, we

may say that work has transformed itself into heat. Using

this mode of expression, the foregoing principle takes the

following form : Work may transform itself into heat, and heat

conversely into work, the quantity of the one bearing always a

fixed proportion to that of the other.

This principle is established by means of many pheno-'mena which have been long recognized, and of late years

has been confirmed by so many experiments of different

kinds, that we may accept it, apart from the circumstance of

its forming a special case of the general mechanical principle

of the Conservation of Energy, as being a principle directly

derived from experience and, observation.




§ '4. Numerical Relation between Heat and Work.

While the mechanicd,! principle asserts that the changes

in the Vis Viva and in the corresponding Work done are

actually equal to each other, the principle which expresses

the relation between Heat and Work is one of Proportion

only. The reason is that heat and work are not measured

on the same scale. Work is measured by the mechanical

imit of the kilogrammetre, whilst the unit of heat, chosen

for convenience of measurement, is That amount of heat

which is required to raise one kilogram of waterfrom 0° to

1" (Centigrade). Hence the relation , existing between heat

and work can be one of proportion only, and the numerical

value must be specially determined.

If this numerical value is so chosen as to give the work

corresponding to an unit of heat, it is called the Mechanical

Equivalent of Heat ; if on the contrary it gives the heat

corresponding, to an unit of work, it is called the Thermal

Equivalent of Work. We shall denote the former by E, and

the latter by -^

The determination of this numerical value is effected in

different ways. It has sometimes been deduced from already

existing data, as was first done on correct principles by

. Mayer (whose method will be further explained hereafter),

although, from the imperfection of the then existing data, his

result must be admitted Hot to have been very exact. At

other times it has been sought to determine the number by

experiments specially made with that view. To the dis-

tinguished English physicist Joule must be assigned the

credit cf having established this value with the greatest cir-

cumspection and care. Some of his experiments, as well as

•determinations carried out at a later date by others, will

more properly find their place after the development of the

theory ;

and we will here confine ourselves to stating those ofJoule's experiments which are the most readily understood,

and at the same time the most certain as to their results.

Joule measured, under various circumstances, the heat

generated by friction, and compared it with the work con-

sumed in producing the friction, for which purpose he

employed descending weights. As accounts of these experi-



EQUIVALENCE OP HEAT AND WORK. 25

ments are given in many books, they need not here be

described ; and it will suffice to state the results as given in

his paper, published in the Phil. Trans., for 1850.

In the first series of experiments, a very extensive one,

water was agitated in a vessel by means of a revolving paddle

wheel, which was so arranged that the whole quantity of

water could not be brought into an equal state of rotation

throughout, but the water, after being set in motion, was

continually checked by striking against fixed blades, which

occasioned numerous eddies, and so produced a large amount

of friction. The result, expressed in English measures^ is

that in order to produce an amount of heat which will raise1 pound of water through 1 degree Fahrenheit, an amount

of work equal to 772'695 foot-pounds must be consumed.

In two other series of experiments quicksilver was agitated

in the same way, and gave a result of 774'083 foot-pounds.

Lastly, in two series of experiments pieces of cast iron were

rubbed against each other under quicksilver, by which the

heat given out was absorbed. The result was 774'987 foot-*

pounds.

Of all his results Joule considered those given by water

as the most accurate; and as he thought that even this

figure should be slightly reduced, to allow for the sound pro-

duced by the motion, he finally gave 772. foot-pounds as the

most probable value for the number sought.

Transforming this to French measures we obtain the

result that. To produce the quantity ofheat required to raise 1

kiiogramme of water through 1 degree Centigrade, work must be

consumed to the amount o/'4!23'55 kilogrammetres. This appears

to be the most trustworthy value among those hitherto

determined, and accordingly we shall henceforward use it as

the mechanical equivalent of heat, and write

J5=423-55 (1).

In most of our calculations it will be sufficiently accurate

to use the even number 424.

§ 5. The Mechanical Unit of Heat.

Having established the principle of the equivalence of

Heat and Work, in consequence of which these two may be




opposed to each other in the same expression, we are often

in the position of having to sum up quantities, in which

heat and work enter as terms to be added together. As,

however, heat and work are measured in different ways, we

cannot in such a case say simply that the quantity is the

sum of the work and the heat, but either that it is the sum

of the heat and of the heat-equivalent of the work, or the sum

of the work and of the workêquivalent of the heat. On

account of this inconvenience Rankine proposed to employ a

different unit for heat, viz. that amount of heat which is

equivalent to an unit of work. This unit may be called simply

the Mechanical Unit of Heat. There is an obstacle to its

general introduction in the circumstance that the unit of

heat hitherto used is a quantity which is closely connected

with the ordinary calorimetric methods (which mainly

depend on the heating of water), so that the reductions

required are slight, and rest on measurements of the most

reliable character ; while the mechanical unit, besides need-

ing the same reductions, also requires the mechanical

equivalent of heat to be known, a requirement as yet only

approximately fulfilled. At the same time, in the theoretical

development of the Mechanical Theory of Heat, in which the

relation between heat and work often occurs, the method of

expressing heat in mechanical units effects siich important

simplifications, that the author has felt himself bound to

drop his former objections to this method, on the occasion of

the present moreconnected exposition

of that theory. Thusin what follows, unless the contrary is expressly stated, it will

be always understood that heat is expressed in mechanical

units.

On this system of measurement the above mentioned

first main Principle of the Mechanical Theory of Heat takes

a yet more precise form, since we may say that heat and its

corresponding work 'are not merely proportional, but equal to

each other.

If later on it is desired to convert a quantity of heat

expressed in mechanical units back again to ordinary heat

units, all that wUl be necessary is to divide the numbergiven in mechanical units by H, the mechanical equivalent of

heat.



EQUIVALENCE OF HEAT AND WOEK. 27

§ 6. Develc^ment of theJtrst main Principle.

Let any body whatever be given, and let its, condition as

to temperature, volume, &c. be assumed to be known. If an

indefinitely small quantity of heat dQ is imparted to this body,the question arises what becomes of it, and what effect it

produces. It may in part serve to increase the amount of

heat actually existing in the body ; in part also, if in conse-

quence of the imparting of this heat the body changes its

condition, and that change includes the overcoming of someforce, it may be absorbed in the work done thereby. If wedenote the total heat existing in the body, or more briefly

the Quantity of Heat of the body, by H, and the indefinitely

small increment of this quantity by dR, and if we put dLfor the indefinitely small quantity of work done, then we can

write

dQ = dH-\-dL (I).

The forces against which the work is done may be

divided into two classes: (1) those which the molecules of

the body exert among themselves, and which are therefore

dependent on the nature of the body itself, and (2) those

which arise from external influences, to which the body is

subjected. According to these two classes of forces, which

have to be overcome, the work done is divided into internal

and external work. If we denote these two quantities by

dJ and d W, we may put

dL =dJ+dW (2),

and then the foregoing equation becomes

dQ = dH+dJ+dW (II).

§ 7. Different conditions of the Quantities J, W, and H.

The internal and external work obey widely different

laws. As regards the internal work it is easy to see that if

a body, starting from any initial condition whatever, goes

through a cycle of changes, and finally returns to its original

condition again, then the internal work done in the whole

process must cancel itself exactly. For if any definite

amount, positive or negative, of internal work remained over

at the end, there must have been produced thereby either an



28 ON THE MECHANICAIi THEORY OF HEAT.

equivalent quantity of external work or a change in the

body's quantity of heat ; and as the same process might be

repeated any number of times it would in the positive case

be possible to create work or heat out of nothing, and in the

negative case to get rid of work or heat without obtaining

any equivalent for it ; both of which results will be at once

admitted to be impossible. If then at every return of the

body to its original condition the internal work done becomes

zero, it follows further that in any alteration whatever of the

body's condition the internal work done can be determined

from its initial and final conditions, without needing to know

the way in which it has passed from one to the other. Forif we suppose the body to be brought successively from the

first condition to the second in several difierent ways, but

always to be brought back to its first condition in exactly

the same way, then the various quantities of internal- work

done in difierent ways in the first set of changes must all be

equivalent to one and the same quantity of internal work done

in the second or return set of changes, which cannot be true

unless they are all equal to each other.

We must therefore assume that the internal forces have an

Ergal, which is a quantity fully determined by the existing

condition of the body at any time, without its being requisite

for us to know how it arrived at that condition. Thus the

internal work done is ascertained by the increment of the

Ergal, which we will call J; and for an indefinitely small

change of the body the differential dJ of the Ergal forms the

expression for the internal work, which agrees with the nota-

tion employed in equations 2 and II.

If we now turn to the external work, we find the state of

things wholly difiêrent. Even when the initial and final

conditions of the body are given, the external work can take

very different forms. To show this by an example, let us

choose for our body a Gas, whose condition is determined by

its temperature t and volume v, and let us denote the initial'

values of these by t^,v^, and the final values by %,v^; let us

also assume that t^>t^, and v^> v^. Now if the change is

carried out in the following way, viz. that the gas is first ex-

panded, at the temperature \, from v^ to v^, and then is

heated, at the volume v^, from <j to t^, then the external

work will consist in overcoming the external pressure which



EQUIVALElfCE OF HEAT AND WORK. 29

corresponds to the temperature f, . On the other hand, if the

change is can-ied out in the following way, viz. that the gas

is first heated, at the volume Vj, from t^ to t^, and is then ex-

panded, at the temperature t^, from v^ to v^, then the ex-

ternalwork will consist in overcoming the external pressure

which corresponds to the temperature t^. Since the latter

pressure is greater than the former, the external work is

greater in the second case than in the first; Lastly, if wesuppose expansion and heating to succeed each other in

stages of any kind, or to take place together according to any

law, we continually obtain fresh pressures, and therewith an

endlessvariety in the quantities of work done with the same

initial and final conditions.

Another simple example is as follows. Let us take a

given quantity of a liquid at temperature t, and transform it

into saturated vapour of the higher temperature t^. This

change can be carried out either by heating the liquid, as a

liquid, to t^, and then vaporizing it; or by vaporizing it at i,

and then heating the vapour to t^, compressing it at the

same time sufficiently to keep it saturated at temperature t^;

or finally by allowing the vaporization to take place at any

intermediate temperature. The external work, which again

shows itself in overcoming the external pressure during the

alteration of volume, has different values in all these different

cases.

The difference in the mode of alteration which, by way

of example, has been thus described for two special classes

of bodies, may be generally expressed by saying that the

body can pass 6y different paths from one condition to the

other.

Another difference, besides this, may come into play. If a

body in changing its condition overcomes an external resist-

ance, the latter may either be so great that the full force of

the body is required to overcome it, or it may be less than this

amount. Let us again take as example a given quantity of

a gas, which at a given temperature and volume possesses

a certain expansive force. If this gas expands, the external

resistance which it has to overcome in so doing must clearly

be smaller than the expansive force, or it would not over-

come it ; but the difference between them may be as small as

we please, and as a limiting case we may assume them to be






sum, the author, in his first Paper on Heat, published in

1850, combined the two under one designation. Following

the same system, we will put

U=H+J.(3),

which changes equation (II) into

dQ = dU+I)W (III).

The function U, first introduced by the author in the above-

mentioned paper, has been since adopted by other writers

on Heat, and as the definition given by him—that starting

from any given initial condition it expresses the sum of the

increment of the heat actually existing and of the heat con-

sumed in internal work—is somewhat long, various attempts

have been made at. a shorter designation. Thomson, in his

paper of 1851*, called it the mechanical energy of a body in

a given state: Kirchoff has given it the name ' Function of

Activity' (Wirkungsfunction)-!": lastly Zeuner, in his 'Grund-

zuge des mechanischen Warmetheorie,' published 1860, has

called the quantity U, when multiplied by the heat-equi-valent of work, the ' Interior Heat ' of the body.

This last name (as remarked in the author's former work

of 1864) does not seem quite to correspond with the meaning

of U; since only one part of this quaptity stands for heat

actually existing in the body, i.e. for vis viva of its molecular

motion, while the other part consists of heat which has been

consumed in doing internal work, and therefore exists as

heat no longer. In his second edition, published 1866, Zeu-

ner has made the alteration of calling tl the Internal Work of

the body. The author however is unable to accept this name

any more than the other, inasmuch as it appears to be just

as objectionably limited on the other side. Of the other

two names, that of Energy, employed by Thomson, appears

very appropriate, since the quantity under consideration cor-

responds exactly with that whichis

denoted by the sameword in Mechanics. In what follows the quantity tT will

therefore be called the Energy of the body.

There still remains one special remark to be made with

reference to the complete determination of the Ergal, and of

* Tram. Royal Soc. of Edinburgh, Vol. xx., p. 475.

t Pogg. Amk Vol. cm., p. 177.



32 ON THE MECHANICAL THEOEY OP HEAT.

the Energy which comprises the Ergal. Since the Ergal

expresses the work which the internal forces must have per-

formed, while the body was passing from any initial condition,

taken as the starting point, to its condition at the moment

under consideration, we can only determine completely, the

value of the Ergal for the present condition of the body,

when we have previously ascertained once for all its initial

condition. If this has not been done, we must conceive

the function which expresses the Ergal as still contain-

ing an indeterminate constant, which depends on the initial

condition. It will be obvious that it is not always necessary

actually to write down this constant, but that we may con-

ceive it as included in the function, so long as this latter is

designated by a general symbol. Similarly we must conceive

another such indeterminate constant as included in the other

symbol which expresses the Energy of the body.

§ 9. Equations for Finite changes of condition-—Cyclical

processes.

If we conceive the equation (III), which relates to an

indefinitely small change of condition, to be integrated for

any given finite change, or for a series of successive finite

changes, the integral of one term can be determined, at once.

Eor the energy U, as mentioned above, depends only on the

condition of the body at the moment, and not on the way in

which it has arrived at that condition. If then we put U^

and CTjj for the initial and final values of U, we may write

/dU=U„-U,.

Hence the equation obtained by integrating (III) may be

written

jdQ= U^- U^+jdTV. (4);

or if we denote by Q and TTthe two integrals j dQ and idWwhich occur in this equation, and which represent respects

ively the total heat imparted tp the body during, the change,

or series of changes, and the external work done, then the

equation will be

Q= U,- U^+W. (4a).



EQUIVALENCE OF HEAT ANB_ WORK. 33

As a special case, we may assume that the body under-

goes a series of changes such that it is finally brought round

to its initial condition. To such a series the author gave the

name of cyclical process. As in this case the initial and

final conditions of the body are the same, U^ becomes equal

to Z7j, and their difference to zero. Hence for a cyclical pro-

cess equations (4) and (4a) become:

jdQ =fdW. (5),

Q=^W (5a).

Thus in a cyclical process the total heat imparted to the

body (i.e. the algebraical sum of all the several quantities of

heat imparted in the course of the cyx;le, which quantities

may be partly positive, partly negative) is simply equal to

the total amount of external work performed.

§10. Total Heat—Latent and Specific Heat.

In former times, when heat was considered to be a sub-

stance, and when it was assumed that this substance might

exist in two different forms, which were distinguished by the

.termsfree and latent, a conception was introduced which was

often made use of in calculations, and which was called the

total heat of the body. By this was understood that quantity

of heat which a body must have taken up in order to pass

from a given initial condition into its present condition, and

which is now contained in it, partly as free, partly as latent

heat. It was supposed that this quantity of heat, if the

initial condition of the body was known, could be completely

determined from its present condition, without taking into

account the way in which that condition had been reached.

Since, however, we have obtained in equation (4a) an

expression for the quantity of heat received by the body in

passing from its initial to its final condition, which expression

contains the external work W, we must conclude that this

quantity of heat, like the external work, depends not only on

the initial and final conditionSj but also on the way in which

the body has passed from the one to the other. The conception

of the total heat as a quantity depending only on the present

c.

3




condition of the body is therefore, under the new theory, no

longer allowable.

The disappearance of heat during certain special changes

of condition, e.g. fusion and vaporization, was formerly ex-

plained, as indicated above, by supposing this heat to pass

into a special form, in which it was no longer sensible to our

touch or to the thermometer, and in which it was therefore

called Latent Heat. This mode of explanation has also been

opposed by the author, who has laid down the principle that

all heat existing in a body is appreciable by the touch and

by the thermometer ; that the heat which disappears under

the above changes of condition exists no longer as heat, buthas been converted into work ; and that the heat which

makes its appearance under the opposite changes (e,g. solidi-

fication and condensation) does not come from any concealed

source, but is newly produced by work done on the body.

Accordingly he. has proposed the term Work-heat as a sub-

stitute for Latent heat in general cases.

This work, into which the heat is converted, and which

in the opposite class of changes produces heat, may be of two

kinds, internal or external. If e.g. a liquid is vaporized, the

cohesion of its molecules must be overcome, and, since the

vapour occupies a larger space than the liquid, the external

pressure must be overcome also. In accordance with these

two divisions of the work we also may divide the total work-

heat, and call the divisions the internal and external work-

heat respectively.That quantity of heat which must be imparted to a body

in order to heat it simply, without making any change in its

density, was formerly known under the general name of free

heat, or more properly, of heat actually existing in the body

a great part of this, however, falls into the same category as

that which was formerly called latent heat, and for which the

term work-Jieat has been proposed. For the heating of a body

involves as a general rule a change in the arrangement of its

molecules, which change produces in general an externally

^perceptible alteration of volume, but still may take place

apart from such alteration. This change of arrangement

requires a certain amount of work, which may be partly

internal, partly external; and in doing this, work-heat is

again consumed. The heat applied to the body thus serves




in part only to increase the heat actually existing, tHe other

part serving as work-heat.

On these principles the author attempted to explain (by

way of example) the unusually great specific heat of water,which is much beyond that either of ice or of steam*: the

assujnption being that of the quantity of heat, which each

receives from without in the process of heating, a larger

portion is consumed in the case of water in diminishing the

cohesion of the particles, and thus serves as work-heat.

From the foregoing it is seen to be necessary that, in

addition' to the various specific heats, which shew how muchheat must be imparted to one unit-weight of a body in order

to warm it through one degree under different circumstances

(e.g. the specific heat of a solid or liquid body under ordinary

atmospheric pressure, and the specific heat of a gas at con-

stant volume or at constant pressure), we must also take into

consideration another quantity which shews hy how much the

heat actually existing in one unit-weight

ofa substance (i.e.

the vis viva of the motion of its ultimate pojrticles) is increased

when the substance is heated through one degree of temperature.

This quantity we will name the body's true heat-capacity.

It would be. advantageous to confine this term 'heat-capa-

city ' (even if the word ' true ' be not prefixed) strictly to the

heat actually existing in the body ; whereas for the total heat

which must be imparted for the purpose of heating it under

any given circumstances, and of which work-heat forms apart, the expression 'specific heat' might be always em-

ployed. As however the term 'heat-capacity' has hitherto

been usually taken to have the same signification as ' specific

heat' it is still necessary, in order to affix to it the above

simplified meaning, to add the epithet ' true,'

§ 11, Expression for the External Work in a particular

case.'

' In equation (III) the external work is denoted generally by

$W. No special assumption is thereby made as to the nature

of the external forces which act on the body, and on which

the external work depends. It is, however, wprth while to

consider one special case which occurs frequently in practice,

• Pogg. Ann, Vol. lxxix., p. 375, and Collection of Memoiri, Vol. l., p. 23.

3—2



36 OIT THE MECHANICAl THEOEY OF HEAT.

and which leads to a very simple determination of the external

work, viz. the case where the only external force acting on the

body, or at least the only force which needs to be referred to

in the determination of the work, is a pressure acting on theexterior surface of the body ; and in which this pressure (as

is always the case with liquid and gaseous bodies, provided

no other forces are acting, and "which may be the case even

with solid bodies) is the same at all points of the surface,

and everywhere normal to it. In this case there is no need,

in order to determine the external work, that we should

consider the body's alterations in form and its expansion in

particular directions, but only its total alteration in volume.As an illustrative case, let us take a cylinder, as shewn

in Fig. 1, closed by an easily moving piston P, and containing

some expansible substance, e.g. a gas, under a .

pressure per unit-area represented by p. Thesection of the cylinder, or the area of the piston,

we may call a. Then the total pressure which

acts on the piston, and which must be overcome

in raising it, is pa. Now if the piston stands

originally at a height h above the bottom of

the cylinder, and is then lifted through an

indefinitely small distance dh, the external work

performed in the lifting will be expressed bythe equation

dW=padh.

But if V be the volume of the gas we have

V = ah, and therefore dv = adh ; whence the above equation

becomes

dW=pdv (6),

This same simple form is assumed by the differential of the

external work for any form

of the body, and any kind

of expansion whatever, as

may be easily shewn as fol-

lows. Let the full line in

Fig. 2 represent the surface

of the body in its original

condition, and the dotted line

its surface after an indefi- Fig. 2.

Kg. 1.



EQUIVALENCK OF SEAT AND WORK. S7

nitely small change of form and volume. Let us consider

Siny element dca of the original surface at the point A.

Let a normal drawn to this element of surface cut the

second. surface at a distance du from the first, where du

is taken as positive if "the position of the second, surface

is outside the space contained within the original surface,

and negative if it is inside. Now let us suppose an inde-

finite number of such normals to be drawn through every

point in the perimeter of the surface-element dm to the

second surface ; there will then be marked out an indefi-

nitely small prismatic space, which, has deo as its base, and

du as its height, and whose volume is therefore expressed by

dadu. This indefinitely small volume forms the part of the

increase of volume of the body corresponding to the element

of surface da. If then we integrate the expression dcodu all

over the surface of the body, we shall obtain the whole

increase in volume, dv, of the body, and if we agree to

express integration over the surface by an integral sign with

suffix CO, we may write

dv- I dudm (7).J ta

Now if, as before, we denote the pressure per unit of

surface by p, the pressure on the element dtu will be pdm.

Therefore the part of the external work, which corresponds

to the element dw, and is described by saying that the

element under the action of the external force pdm is pushed

outwards at right angles through the distance du, will beexpressed by the i^rodact pdm du. Integrating this over the

whole surface, we obtain for the total external work,

dW = I pdudm.J ia'

As p is equal over the whole surface, the equation may be

written

dW=p\ dudm,J ta

or, by equation (7),

dW=pdv,

which is the same as equation (6) given above.





(39)

CHAPTER II.

ON PEBFECT GASES.

§ 1. The Gaseous condition of bodies.

Among the laws which characterize bodies in the gaseous

condition the foremost place must be given to those of

Mariotte and Gay Lussac, which may be expressed together

in a single equation as follows. Given a unit-weight of a

gas, which at freezing temperature, and under any standard

pressure p^ {e.g. that of the atmosphere) has the volume «„j

then if p and v be its pressure and volume at any tempera-

ture t (in Ceritigrade measure) the following equation will

hold:

pv=-p^v^{\ + a.t) ; (1),

wherein the quantity a, which is usually termed the coeffi-

cient of expansion, although it really relates to the change

of pressure as well as the change of volume, has one and the

same value for every kind of gas.

Regnault has indeed recently proved by careful experi-

ment that these laws are not strictly accurate; but the

deviations are for permanent gases very small, and become of

importance only for gases which are capable of condensation.

It seems to follow that the laws are the more nearly exact,

the further a gas is removed, as to pressure and temperature,^from its point of condensation. Since for permanent gases >

under ordinary conditions the exactness of the law is already,

so great, that for most purposes of research it may be taken

as perfect, we may imagine for every gas an ultimate condi-

tion, in which the exactness is really perfect; and in what

follows we will assume this, ideal condition to be actually




reached, calling for brevity's sake all gases, in which this is

assumed to hold, Perfect Gases.

As however the quantity a, according to Eegnault's deter-

minations, is not absolutely the same for all the gases which

have been examined, and has also somewhat different values

for one and the same gas under different conditions, the

question arises, what value we are to assign to a in the case

of perfect gases, in which such differences can no longer

appear. Here we must refer to the values of a which have

been found to be correct for various permanent gases. By

experiments made on the system of increasing the pressure

while keeping the volume constant, Regnault found the fol-

lowing numbers to be correct for various permanent gases

Atmospheric Air 0003665.

Hydrogen 0003667.

Nitrogen 0003668.

Carbonic Oxide' 0-003667.

The differences here are so small, that it is of little im-

portance what choice we make; but as it was with atmo-

spheric air that Regnault made the greatest number of

experiments, and as Magnus was led in • his researches to a

precisely similar result, it appears most fitting to select the

number 0003665.

Regnault, however, by experiments made on the other

system of keeping the pressure constant and increasing the

volume, has obtaitied a somewhat different value for a in the

case of atmospheric, air, viz. 0'003670. He has further

observed that rarefied air gives a somewhat smaller, and com-

pressed air a somewhat larger, coefficient of expansion than

air of ordinary density. This latter circumstance has led

some physicists to the conclusion that, as rarefied air is nearer

to the perfect gaseous condition than air of ordinary density,

we ought to assume for perfect gases a smaller value than

0003665. Against this it may be urged, that Regnaultobserved no such dependence of the coefficient of expansion

on the density in the case of hydrogen, but after increasing

the density threefold obtained exactly the same value as

before ; and that he also fou©d that hydrogen, in its devia-

tion from the laws of Mariotte and Gay Lussac, acts altogether

differently, and ior the, most 'part in exactly the opposite





42 ox THE MECHANICAL THEORY OF HEAT,

fully explained further on. I'aking the value of a given

in equation (2) we obtain

<^ =l

=

M(7).

T=273 + t]

§ 2. Approadmate Principle as to Heat absorbed ly Gases.

In an experiment of Gay Lussac's, a vessel filled with air

was put in communication with an exhausted receiver of

equal size, so that half the air from the one passed over into

the other. On measuring the temperature of each half, and

comparing it with the original temperature, he found that

the air which had passed over had become heated, and the

air which remained behind had become cooled, to exactly the

same degree; so that the mean temperature was the same

after the expansion as before. He thus proved that in this

kind of expansion, in which no external work was done, no

loss of heat took place. Joule, and after him Regnault,

carried out similar experiments with greater care, and bothwere led to the same result.

The principle here involved may also be deduced, without

reference to special experiments, from certain properties of

gases otherwise ascertained, and its accuracy may thus be

checked. Gases shew so marked a regularity in their beha-

viour (especially in the relation between volume, pressure,

and temperature, expressed by the law of Mariotte and Gay

Lussac), that we are thereby led to the supposition that

the mutual action between the molecules, which goes on in,

the interior of solid and liquid bodies, is absent in the case

of gases ; so that heat, which in the former cases has to over-

come the internal resistances, as well as the external pressure,

in order to produce expansion, in the case of gases has to do

with external pressure alone. If this be so, then, if a gas

expands at constant temperature, only so much heat canthereby be absorbed as is required for doing the external

work. Again, we cannot suppose that the total amount of

heat actually existing in the body is greater after it has

expanded at constant temperature than before. On these

assumptions we obtain the following principle ; a permanent

gas, if it expands at a constant temperature, absorbs only






The condition of the gas is completely determined, when

its temperature and volume are known ; or it may be deter-

mined by its temperature and pressure, or by its volume and

pressure. We will at present choose the first-named quan-

tities, temperature and volume, to determine the condition,

and accordingly treat T and v as the independent variables,

on which all other quantities relating to the condition of the

gas depend. If then we regard the energy U of the gas as

being also a function of these two variables, we may write

drdv

v^hence equation (IV) becomes

dU ,^. fdU

^^-^'^-îf^^y^(«)

This equation, which in the above form holds not only

for a gas, but for any body whose condition is determined'

by its temperature and volume, may be considerably simpli-

fied for gaseous bodies, on account of their peculiar proper-

ties.

The quantity of heat, which a gas must absorb in ex-

panding at constant temperature through a volume dv, is

generally denoted by -r^ dv. As by the approximate assump-

tion of the last Section this heat is equal to the work donein the expansion, which is expressed by pdv, we have the

equation

dQ, . dQ

But from equation (8)

dQ^dV^dv dv ^

'

hence from the last two equations we obtain

f-» (»)•



ON PERFECT GASES. 45

Hence we conclude that in a perfect gas the energy U is

independent of the volume, and can only be a function of

the temperature.77r

If in equation (8) we put -5- = 0, and substitute for

j-~, the symbol C,, it becomes

dQ= C,dT+pdv (10).

From the form of this equation we see that C„ denotes

the Specific Heat of the Oas at constant volwme, since GjiT

expresses the quantity of heat which must be imparted tothe gas in order to heat it from 2' to T+dT, when du is

equal to zero. As this Specific Heat = -tj,, i.e. is the differ-

ential coefl&cient with respect to temperature of a function of

the temperature only, it can itself also be only a function of

temperature.

In equation (10) all the three quantities T, v, and p are

found ; but since by equation (6) pv — Rt, it is easy to

eliminate one of them ; and by eliminating each in succession

we obtain three different forms of the equation.

Eliminating p we obtain,

dQ'=C,dT+ -^d'B (11).

7?7'

Again, to eliminate v we put v =^— ; whence we have

dv= — dT^—T- ip.

If we substitute this value of dv in equation (10), and

then combine the two terms of the equation which contain

dT, we obtain

dQ = {C, + B)dT-?^dp (12).

Lastly, to eliminate T, we obtain from equation (6), by

differentiation,

-,_vdp+pdii






investigation. Regnault tested atmospheric air at pressures

from 1 to 12 atmospheres, and hydrogen at from 1 to 9 at-

mospheres, but could detect no difference in their specific'

heats. He tested them also at different temperatures, viz.

between - 30° and + 10°, between 0° and 100°, and between0° and 200° ; and here also he found the specific heatt always

the same *. The result of his experiments may thus be ex-

pressed by saying that, within the limits of pressure and

temperature to which his observations extended, the specific

heat of permanent gases was found to be constant.

It is true that these direct explanatory researches were

confined to the specific heat at constant pressure; but therewill be little scruple raised as to assuming the same to be

correct for the other specific heat, which by equation (14)

differs from the former only by the constant B. Accordingly

in what follows we shall treat the two, specific heats, at least

for perfect gases, as being constant quantities.

By help of equation (14) we may transform the three

equations (11), (12) and (13), which express the first main

principle of the Mechanical Theory of Heat as applied to

gases, in such a way that they rnay contain, instead of the

Specific Heat at constant volume, the Specific Heat at con-

stant density ; which may perhaps appear more suitable, since

the latter, as being determined by direct observation, ought

to be used more frequently than the former. The resulting

equations are

.(15).

V

dQ = C,dT-^^dp

dQ = ^^^vdp + ^pdv

Lastly, we may introduce bothSpecific Heats into the

equations, and eUminate B, by which means the resulting

* The numbers obtained for atmospheric air (Be/, des Exp. Vol. il., p. 108)

are as follows in ordiuary heat units

between - 30» and + 10» 0'23771,

„ 00 „ I'OO" 0-23741,

„ O" „,

208» 0-23751,

which may be taken as practically the same.



48 ON THE MECHANICAL THEORY OF HEAT,

equations become symmetrical as to^ and v, as follows :

.(16).

G

In the above equations the speciiic heats are expressed

in mechanical units. If we wish to express them in ordinary-

heat units, we haveonly to divide these values by the

Mechanical Equivalent of Heat. Thus if we denote the

specific heats, as expressed in ordinary heat units, by c, and

Cj,, we may put

«.= §. ^ =^ (17)-

Applying these equations to equation (14), and dividing

by E,we

have

c. = c,+ |, (18).

§ 5. Relation between the two Specific Heats, and its

application to calculate the Mechanical Equivalent of Heat.

If a system of Sound-waves spreads itself through any gas,

e.g. atmospheric air, the gas becomes in turn condensed and

rarefied; and the velocity with which the sound spreads

depends, as was seen by Newton, on the nature of the changes

of pressure produced by these changes of density. For very

small changes of density and pressure the relation between

the two is expressed by the differential coefficient of the

pressure with respect to the density, or (if the density, i.e. the

weight of a unit of volume, is denoted by p) by the differential

coefficient -y-. Applying this principle we obtain for the

velocity of sound, which we will call u, the following equa-

tion

« = V;4 <'^''

in which g represents the. accelerating force of gravity.



ON PERFECT GASES. 49'

Now in order to determine the value of the diffe'rential

coefficient-J-

Newton used the law of Mariotte* according to

which pressure and density are proportional to each other.

pdp-pdp

He therefore put - = constant, whence by differentiation

and therefore

whence (19) becomes

.- =»

t-l m:

«-\/«:- .(21).

The velocity calculated by this formula did not however agree

with experiment, and the reason of this divergence, after it

had been long sought for in vain, was at last discovered byLaplace.

The law of Mariotte in fact holds only if the change of

density takes place at constant temperature. But in sound

vibrations this is not the case, since in every condensation

a heating of the air takes place, and in every rarefaction g,

cooling. Accordingly at each condensation the pressure is

increased, and at each rarefaction diminished, to a greater

extent than accords with Mariotte's law. The question now

arises how, under these circumstances, can the value jof -y-

dp

be determined.

Since the condensations and rarefactions follow each other

with great rapidity, the exchange of heat that can take place

during each short period between the condensed and rarefied

parts of the gas must be very small. Neglecting this, wehave to do with a change of density, in which the quantity of

gas under consideration receives no heat and gives forth none;

and we may thus, in applying to this case the differential equa-

* Tkis law is commonly known in England as ' Boyle's law,' as being

originally due to Boyle. (Translator.)

c. 4



So ON THE MECHANICAL THEORY OF HEAT.

tions of the last section, put dQ = 0, Hence, e.g. from the last

Qf equations (16),, we olstain:

or G^vdp + Oj,pdv = 0.

Now, since the volume v of one unit of weight 18 the re-

ciprocal of the density, we may put v = -, and,therefore

r

dv =.

—j^ ; whence the equation becomes

p p

or # = £p2 (22).

This value of the Differential Coefficient -^ differs from thatdp

deduced from Mariotte's law, and given in (20), by containing

as factor the ratio of the two Specific Heats. If for simplicity

we put

^ =1

(23),

the last equation becomes

^ = hP (24),dp p

Substituting this value of -^ in equation (19), we get instead

of (21)

« = \/%^ (25)v

From this equation the' velocity of sound u can be calculated

if A is known ; or, on the other hand, if the velocity of sound

is known by experiment, we can apply the equation to calcu-

late k, changing it firs': into the form

h^"^^ (26).gp

^ J



ON PEBFECT GASES. 51

The velocity of sound iu air has been several times deter-

mined with great care by various physicists, -whose results

agree with each other very closely. According to the experi-

ments of Bravais and Martens* the velocity at freezing tem-

perature is 332"4 m. per second (10906 feet). We will in-

Siert this value in equation (26). We may also give g its

recognized value 9'809 m. (32-2 feet). To determine the

quotient - we may give the pressure p any value we please,

but we must then assign to the density p the value corre-

sponding to that pressure. We will assume p to be the

pressure of 1 atmosphere. This must be expressed in the

fonnula by the amount of weight supported per unit of

surface. As' this weight is equal to that of a column of

quicksilver, whose base is 1 sq. m. and height 760 mm., and

which therefore has a volume of 760 cubic decimetres, and as,

according to Regnault, the Specific Weight of quicksilver at

0°, as compared with water at 4°, is 13"696, we obtain

p = l atmosphere = 760 x 13"596 = 10333 kg. per sq. metre.

Lastly, p is the weight of a cubic metre of air under the

assumed pressure of 1 atmosphere and at temperature 0°,

which, according to Regnault, is 1'2932 kg. Substituting

these values in equation (26) we obtain

(332-4)- X 1-2932,

*~ 9-809x10333 ~*^"-

Ha-ving thus determined the quantity h for atmospheric

air, we can now use equation (18) to calculate the quantity E,

i.e. the Mechanical Equivalent of Heat, as was first done by

Mayer. For we have from (18)

and, if we again denote by k the quotient -s, which is the

» Ann. de Chim. iii., 13, 5; and Fogg. Am. Vol. lxvi., p. 351.

4—2




C Csame as -^ , and accordingly substitute -^ for c„ we have

^-of%-:''^-

Here we may substitute for k its value 1"410 just found,

and for c^ its value as given by Regnault, 02375. It then

remains to determine B, or ^-^°. To do this, let us agam

take Pn as the pressure of 1 atmosphere, which, as seen above,

is equal to 10333, and we then have for v„ the volume in

cubic metres of 1 kg. of air under the, above pressure of

1 atmosphere and at temperature 0", which according to

Regnault is 0'7733. Lastly we have already assumed the

value of a to be 273. The value of B for atmospheric air

will therefore be given by the equation

P _ 10333 X 07733 __„„273

Substituting these values for k, c^, and B in equation (27)

we obtain

1-410 X 29-27 _^ ~ 0-410 X 0-2375

~*'^'^^-

This figure agrees very closely with that determined by

Joule from the friction of water, viz. 423-55. In fact it

must be admitted that the agreement is more close than,considering the degree of uncertainty as to the data used in

the calculation, we could have had any right to expect; so

that chance must have assisted in some degree to produce it.

In any case, however, the agreement forms a striking con-

firmation of the equations deduced for permanent gases.

§ 6. Various Formulce relating to the Specific Heats of

If in equation (18), p. 48, we consider the quantity âsknown, we may apply that equation to calculate the specific

heat at constant volume from that at constant pressure, which

is known from experiment. This application is of special

importance, because the method of deducing the ratio of the




two specific heats from the velocity of sound is only prac-

ticable in the case of the very few gases for which that

velocity has been experimentally determined. For all others,

equation(18)

offers the only means as yet discovered of

calculating the specific heat at constant volume from that

at constant pressure.

It must here be observed that equation (18) is exactly

true only for perfect gases, although it gives at least approxi-

mate results for other gases. The circumstance has also to

be considered, that the determination of the specific heat of

a gas at constant pressure is the more difficult, and therefore

the value determined the less reliable, in proportion as thegas is less permanent in its character, and thus diverges

more widely in its behaviour from the laws of a perfect gas

therefore, as there is no need to seek in our calculations a

greater accuracy than the experimental values themselves

can possibly possess, we may treat the mode of calculation

employed as sufficiently complete for our purpose.

Accordingly we begin by putting equation (18) in the form

o. = c,-§ (28).

Here for E we shall use the value 423'55. S is determined

by equation (4)

It =a

where p„v„ are the pressure and volume at the temperature of

freezing. Should it be difficult to make observations on the

gas at this temperature (as is the case with many vapours)

we may also, by equation (6), give B the value

i?=5 (29),

where p, v, and T are any three corresponding values of

pressure, volume, and absolute temperature.

This quantity B, as already observed, is only so far depen-

dent on the nature of the gas, that it is inversely proportional

to its specific gravity. For if we denote by v' the volume

of. a unit of weight of air at temperature T and pressure p.



54 O'S THE MECHANICAL THEORY OF HEAT.

and by R' the corresponding value of R, we have

"' ji •

Combining this with equation (29),

B = R'^,V

But -, is the ratio of the volumes of equal weights of the two

gases, and is therefore the reciprocal of the ratio of the

weights of equal volumes, which ratio is called the Specific

Gravity of the gas, as compared with common air. If we

call this specific gravity d, the last equation becomes

^ =f (^«)-

Substituting this value of B in (28) we obtain

'^-'^'-M(^1)-

The quantity here denoted by R', i.e. the Value of R for

atmospheric air, has been already determined in § 5 to be

equal to 29-27. Hence further,

whence the equation, which serves to determine the Specific

Heat at constant volume, takes this simple form :

0-0691''« = ". ^— (32).

If in the next place we apply this equation to the case of air,

for which d=l, and for the sake of distinction denote byaccented letters the two specific heats for air, we get thefollowing equation

c' =c;- 00691 (33),




and substituting for c'^ its value according to Kegnault,

which is 02375, we obtain the result

c'.= 0-2375 -0-0691 = 0-1684 (34.)*

!For the other gases the equation may be given in the

following form

^^^^-0:0691^3.^^

which, as will be seen later, is specially convenient for the

application of the values given by Eegnault for specific

heats at constant pressure.

The specific heats denoted by c^ and c„ relate to a unit

of weight of the gas, and have for unit the ordinary unit of

heat, i.e. the quantity of heat required to raise a unit of

weight of water from the temperature 0° to 1". We maythus say that the gas, in relation to the heat which it

requires to raise its temperature either at constant pressure

or constant volume, is referred as regards weight to the

standard of water., With gases however it is desirable to refer to the

standard of air as regards volume ; i.e. so to determine the

specific heat, as to compare the quantity of heat, which

the gas Requires to raise its temperature through 1°, with the

quantity of heat which an equal volume of air; taken at the

same temperature and pressure, requires to raise its tempe-

rature to the same extent. We may use this kind of

comparison in the case of both the specific heats, inasmuch

as we assume in the one case that both the gas under con-

sideration and the atmospheric air are heated at constant

pressure, and in the other that they are both heated at

constant volume. The specific heats thus determined may

be denoted by 7^ and 7^.

As we denote by v the volume which a unit weight of gas

assumes at a given pressure and temperature, the quantity ofheat, which a unit-volume of the gas absorbs at constant

pressure in being heated through 1°, will be expressed by -^

• It will be Been that cj and e,' fulfil the condition fonnd , above for

perfect gases; — = 1-410. {Translator.)




c

or iu the case of atmospheric air by -f The specific heat

7^ is found by dividing the former quantity by the latter, or,

^^ = ^x-^;=%x^ = ^cZ (36).

Similarly % = ^d (37).c.

In the first of these two equations we may give to c'^ its

value as found by Regnault, 0'2375 ; the equation thenbecomes

^ =0^ (^«)-

In the second we may put for c'„, according to (34)^ the

value 0"1684<, and for c, the expression given in (35); whence

we have

c„d- 00691 ,„Q>

§ 7. Numerical Calculation of the Specific Heat at con-

stant Volume.

The formulae developed in the last section have been

applied by the author to calculate from the values which

Eegnault has determined by his researches for the Specific

Heat at constant Pressure of a large number of gases and

vapours, the corresponding values of the Specific Heat at

constant Volume. In so doing he has in some sort recal-

culated one of the two series of numbers given by Regnault

himself; who has expressed the Specific Heat at constant

Pressure in two different ways, and has brought together the

resulting numbers in two series, one of which is superscribed' en poids,' and the other ' en volume.' The first series con-

tains the values which result, if the gases in question are

compared weight by weight with water, in relation to the

quantity .of heat. required to warm them through 1°; in other

words, the values of the quantities denoted above by Cp. Thenumbers in the second series are simply obtained from those




in the first by multiplying them by the corresponding specific

gravity, i. e. they are the values of the product Cpd.

These latter numbers were no doubt those most easily

• calculated from the observed values of c^,; but their signi-

fication is somewhat complicated. With them the quan-

tity of heat has for its unit the ordinary unit of heat, whilst

the volume to which they refer is that which a unit-weight

of atmospheric air assumes, when under the same tempera-

ture and pressure as the gas under consideration. The tedious-

ness of the verbal description thus required makes the

numbers troublesome to understand and to apply ; moreover

this mode of expressing the Specific Heat of gases has beenused, so far as the author knows, by no previous writer. In

considering gases with reference to volume, it has in all other

cases been custoinary to compare the quantity of heat, which

a given gas requires to raise its temperature through 1°, with

the quantity of heat which an equal volume of atmospheric

air requires under the same conditions for the same purpose,

or, as briefly expressed above, by comparing the gas, volume

for volv/me, with air. The numbers thus obtained are re-

markable for their simplicity, and allow the laws which

hold as to the specific heats of the gas to be treated with

special clearness.

It will therefore, the author believes, be found an ad-

vantage that he has calculated, from the values given by

Eegnault under the heading 'en volume' for the product

Gj,d, the values of the quantity 7^,, defined above. All thatwas required for this, by (38), was to divide the values of c^d

by 0-2375.

He has further calculated the values of c„ and 7,; calcula-

tions which by equations (35) and (39) could be very simply

performed, by taking fi:om the values of the product c^d the

number 0'0691, and dividing the remainder by d, or by

01684, respectively.

The numerical values thus calculated are brought together

in the annexed table, in which the different columns have the

following signification

Column I. gives the name of the gas.

Column II. gives the Chemical composition, and this

expressed in such a way that the diminution of volume pro-



S8 ON THE MECHANICAL THEORY OF HEAT.

duced by the combination can be immediately observed. For

in each case those volumes of the simple gas are given, which

must combine in order to give Two Volumes of the compound

gas. Thus we assume for Carbon, as a gas, such an hypothe-

tical volume as we must assume, in order to say that one

volume ofCarbon unites with one volume of Oxygen to makeCarbonic Oxide, or, with two volumes to make Carbonic

acid. Again, when, e.g. Alcohol is denoted in the Table by

CjjHgO, this means that two volumes of the hypothetical car-

bon gas, six volumes of Hydrogen, and one volume of Oxygen,

make up together two volumes of Alcoholic vapour. For

sulphur-gas the specific gravity used to determine its volumeis that found by Sainte-Claire Deville and Troost for very

high temperatures, viz. 2"23. In the five last combinations

in the Table, which contain Silicon, Phosphorus, Arsenic,

Titanium, and Tin, these simple elements are denoted by

their ordinary chemical signs, without reference to their

volumes in the gaseous condition, because the gaseous,

volumes of these elements are partly still unknown, partly

hampered with certain irregularities not yet thoroughly cleared

up.

Column III. gives the Density of the gas, using the values

given by Kegnault.

Column IV. gives the Specific Heat at constant Pressure

as compared, weight for weight, with water, or in other words

referred to a unit-weight of the gas and expressed in ordinaryunits of heat. These are the numbers given by Regnault

under the heading ' en poids.'

Column V. gives the Specific Heat at constant Pressure

compared, volume for volume, with air, calculated by divid-

ing by 0'2375 the numbers given by Regnault under the

heading ' en volume.'

Column VI. gives the Specific Heat at constant Volumecompared, weight for weight, with water, calculated by equa-

tion (35).

Column VII. gives the Specific Heat at constant Volumecompared, volume for volume, with air, calculated by equation

(39).





60 ON THE MECHANICA.L THEORY OF HEAT.

§ 8. Integration of the Differential Equations which ex-

press the first main Principle in the case of Oases.

The differential equations deduced in sections 3 and 4,which

in various forms express the first main principle of the Mechani-

cal Theory of Heat in the case of gases, are not immediately

integrable, as can be seen by inspection; and must therefore

be treated after the method developed in § 3 of the Introduc-

tion. In other words, the integration becomes possible as

soon as we subject the variables occurring in the equation to

some one condition, thus determining the path of the change

ofcondition of the body. We shall here give only two very

simple examples of the process, the results of which are

important for our further investigations.

Example 1. The gas changes its volume at Constant

Pressure, and the quantity of heat required for such change is

known.

In this case we select from the above equations one which

contains

p and v as independent variables, e.g. the last ofEquations (15), which is

dQ =^^vdp + ^piv.

As the pressure p is to be constant, we put p=p^, and

dp = 0; the equation then becomes

dQ = -^p^dv,

which gives on integration (if we call v^ the original value

oi v)

<3 = §Fi(«-«i) (40).

Example 2. The gas changes its volume at ConstantTemperature, and the quantity of heat required for suchchange is known.

In this case we select an equation which contains y and vas independent variables, e.g. Equation (11), which is

dQ = C,dT +— dv.V




As Tis to be constant, we put T= T^, and dT= ; whencewe have

1 V

Integrating,

V

.Q = RT>g^ (41).

Hence is_ derived the Principle that if a Gas changes its

volume without change of temperature, the quantities of heat

absorbed or given off form an arithmetical series, while the

volumesform a geometrical series.

Again, if we put for R its value ^^ , we have

Q=p,v,log^ (42).

_If we suppose this equation.to refer, not directly to a unit

weight of the gas, but to a quantity of it such that at pressure

p it assumes a volume v^, and then suppose that this volumechanges under constant temperature to v, then the equation

contains nothing which depends on the special nature of the

gas. Therefore the quantity of heat absorbed is independent

of the nature of the gas. Further, it does not depend on the

temperature, but only on the pressure, being proportional to

the original pressure.

Another application of the differential equations deduced

in sections 3 and 4 consists in making some assumption as

to the heat to be imparted to the gas during its change

of condition, and then enquiring what course the change of

condition will take under such circumstances. The simplestand at the same time most important assumption of this kind

is that no heat whatever is imparted to or taken from the gas

during its change of condition. For this purpose we mayimagine the gas confined in a vessel impermeable to heat, or

that the change is so rapid that no appreciable heat can pass

to or ffom the gas in the time.




On this assumption we must put dQ = 0. Let us do

this for the three Equations (16). Then the first of these

becomes

C,dT+{G^-G:)^dv = 0.

GDividing by T x (7„ and as before denoting jf by h, we have

f+(*-i)?=o.

Integrating,

log T+(k-l)logv = Const.

or Tv^^ = Const.

If Tj, Vj are the original values of T, v, we may eliminate the

Constant, and obtain

.(43).

If this equation be applied for example to atmospheric air,

then, writing k= 1"410, we can easily calculate the change of

temperature which corresponds to any given change of

volume. If e.g. we assume a certain quantity of air to be

taken at freezing temperature and at any pressure whatever,

and to be compressed, either in a vessel impermeable to

heat, or with great rapidity, to half its volume, then Tj = 273

(absolute temperature) and — = 2 ; hence the equation be-

comes

^'=2»^'» = 1-329,

whence T= 273 x 1-329 = 363,

or if t be the temperature measured in degrees above freezing

point,

«= T- 273 = 90°.

If a similar calculation is made for the compression of the

gas to J and -^ of its original volume, results are obtained,



ON PEEFECT GASES. ^3

which, combined with the former, are presented in the follow-

ing Table.

Value of

V



64 ON THE MECHANICAL THEORY ÔF HEAT.

§ 9. Determination of the External Work done during

the change ofvolwme of a gas.

There is one quantity connected witli the expansion of a

gas which still requires to be specially considered, viz. the

External Work done in the process. The element of this

work, as determined in Equation (6), Ch. I, is

dW =pdv.

This work may be very clearly set forth by a graphic

representation. We will adopt a rectangular system of co-or-

dinates, in which the abscissa represents the volume v, and the

ordiiiate the pressure p. Ifwe now suppose p to be expressed

as a function of v, say p =f (v), then this equation is the

equation to a curve, whose ordinates express the values of pr

corresponding to the different values of v, and which for

brevity we will call the Pressure-curve. In Fig. 3 let rs be

this curve, so that, if oe repre-

sent the volume v existing ata certain instant, the ordinate

ef drawn at e will represent

the pressure at the same in-

stant. If further eg represent

an indefinitely small element

of volume dv, and the ordinate

gh is drawn at g, then we shall

have an indefinitely small para-

lellogram efhg, whose area re-

presents the external work

done in an indefinitely small

expansion of the body ; and

which differs from the product pdv only by an indefinitely

small quantity of the second order, which may be neglected.

The same holds for any other indefinitely smaU expansion;

and hence in the case of a finite expansion (say from the

volume Vj, represented by the abscissa oa, to that otv^, repre-

sented by the abscissa oc) the external work, for which wehave the equation

Fig. 3.

W = I pdv , .(46),






the value^' , where 'p^^ is any value obtained for the conr

stant in the above equation ; we then get from (46)

^=PA£''7=i'iîôg^^ (*®)-

We observe that this value of TT coincides with that given

in equation (42) for Q ; the reason for this being that the gas,

while expanding at constant temperature, absorbs only so

much heat as is required for the external work.

Joule has employed the equation(48)

in one of his deter-

minations of the Mechanical Equivalent of Heat. For this

purpose he forced atmospheric air into a strong receiver, up

to ten or twenty times its normal density. The receiver and

pump were meantime kept under water, so that all the heat

which was developed in pumping could be measured in the

water. The apparatus is represented in Fig. 6, in which H is

the receiver, and G the pump. The vessel G, as will be

easily understood, was used for the drying of the air, and thevessel with the spiral tube served to give to the air, before its

entrance into the pump, an exactly known temperature.

From the total quantity of heat given in the calorimeter

Joule subtracted the part due to the friction of the pump,

the amount of which he determined by working the pumpfor exactly the same length of time, and under the same

mean pressure, but without allowing the entrance of air, and

then observing the heat produced. The remainder, after this

was subtracted, he took as being the quantity of heat de-

veloped by the compression of the air ; and this he compared

with the work required for the compression as given by equa-

tion (48). By this means he obtained as the mean of two

series of experiments the value of 444 kilogrammetres as the

Mechanical Equivalent of Heat.

This value, it must be admitted, does not agree very wellwith the value 424 obtained by the friction of water ; the

reason of which is probably to be found in the far larger

sources of erroi: attending experiments on air. Nevertheless

at that time, when the fact that the work required for

developing a given quantity of heat was equal under all cir-

cumstances was not yet placed on a firm basis, the agreement






In this case the relation between pressure and volume is

given by equation (45), viz.:

Pi \vj

'

"The curve of pressure corresponding to this equation

(Fig. 7) falls more steeply than that delineated in Fig. 5.

ilankine has given to this

special class of pressure-

curves, which correspond to

the case of expansion within

an envelope impermeable to

heat, the name of Adiabatic

curves (from Sia/3alveiv, to

pass through). On the

other hand Gibbs {Trans.

Connecticut Academy, vol.

II. p. 309) has proposed to

namethemIsentropiccurves,

because in this kind of ex-

pansion the Entropy, a quan-Fig. 7.

tity which will be discussed further on, remains constant. This

latter form of nomenclature is the one which the author pro-

poses to adopt, since it is both usual and advantageous to

designate curves of this kind according to that quantity

which remains constant during the action that takes place.

To effect the integration in this case, we may put, accord-ing to the above equation,

I>=FiK

whence (46) becomes

(49).







SECOND MAIN PRINCIPLE. 7l

From henceforward let the body be compressed, so as to

bring it back, to its original volume. Let the compression

first take place at the constant temperature T^, for which

purpose we may suppose that the body is connected with a

body K^ at temperature T^, acting as a reservoir of heat, and

that it gives up to K^ just so much heat as suffices to keep

itself also at temperature T^. The pressure-curve correspond-

ing to this compression is again an isothermal curve, and in

the special case of a perfect gas is another equilateral hyper-

bola, of which we obtain the portion cd during the reduction

of volume to oh=v^.

Finally, let the last compression, which brings the varia-ble body back to its initial volume, take place without

the presence of the body K^, so that the temperature rises,

and the pressure follows the line of an isentropic curve.

We wUl assume that the volume oh = v^, up to which the

compression went on according to the first mode, is so chosen,

that the compression which begins from this volume and

continues to volume oe = v^ is just sufficient to raise the

temperature again from T^ to T^. If then the initial tem-

perature is thus regained at the same time as the initial

volume, the pressure must also return to its initial value, and

the last curve of pressure must therefore exactly hit the

point a. When the body is thus brought back again to the

original condition, expressed by the point a, the cyclical pro-

cess is complete.

§ 2. Result of tJie Cyclical Process.

During the two expansions which take place in the course

<jf the cyclical process the ex-

ternal pressure must be over-

come, and therefore external

work must be performed

whereas conversely during the

compressions external workis absorbed to perform them.

These quantities of work are

given directly by the figure,

which is here reproduced.°' ^'

The work performed during

the expansion ab is represented by the quadrangle eabf, and




that perfortned during the expansion be by the quadrangle

fbcg. Again, the work absorbed for the compression cd is

represented by the quadrangle cfcdh, and that absorbed for

the compression da by the quadrangle hdae. The two latterquantities, on account of the lower temperature which obtains

during the compression, are smaller than the two former;

and, if we subtract them from these, there remains an over-

plus of external work performed, which is represented by the

quadrangle abed, and which we will call W.

To the external work thus gained must correspond, ac-

cording to equation (5a) of Chapter I., a quantity Q, equal to

it in value, which is required for its production. Now the

variable body, during the first expansion, expressed by ab,

which took place in connection with the body if,, received

from this latter a certain quantity of heat, which we may call

Q, ; and again during the first compression, expressed by cd,

which took place in connection with the body K^, it im-

parted to this latter a certain quantity of heat, which may be

called Qj. During the second expansion 6c and the second

compression da the body neither imparted nor received heat.

Now, since in the course of the whole cyclical process a certain

quantity of heat Q is. absorbed in work, it follows that the

quantity of heat Q,, received by the variable body, is larger

than the quantity of heat Q^ which it gives out, so that the

difference Q^ — Q^ is equal to Q.

We may accordingly put

Q. = Q,+ Q (1),

and can tben distinguish in the quantity of heat Q,, which

the variable body has drawn from the body iT,, two parts, of

which one Q is converted into work, whilst the other Q, is

given back as heat into the body K^. Since in all the other

relations of the body the original condition is restored at the

end of the cyclical process, and accordingly every variation

which takes place at one part of the process is counter-

balanced by an equal and opposite variation which takes

place at some other part of the process, we may finally de-

scribe the result of the cyclical process in the following terms

The one quantity of heat Q, derivedfrom the bodyK, is trans-

formed into work, and the other quantity Q^ has passed over

from the hotter body Kj into the colder K^.



SECOXD MAIN PRINCIPLE. 73'

The whole of the cyclical process just described may also

be supposed to take place in tê reverse order. If we again

begin with the conditions represented by the point a, in

which the variable body has the volume «, and the tempera-ture Tj, we may suppose that it first expands, without any

heat being imparted to it, to the volume v^, thus describing

the curve ad, in which its temperature sinks from T^ to T^;

that it is then connected with the body K^, and expands at

constant temperature T^ from v^ to Fj, describing the curve

dc, during which it draws heat from the body K^; that it

then, without parting with its heat, is compressed from V^ to

F,, describing the curve ch, during which its temperature rises

from Tjj to T^ ; finally that it is connected with the body K^,

at the constant temperature T^, and whilst imparting its heat

to K^ is again compressed from Fj to the initial volume v^,

describing the ctirve ha.

In this reversed process the quantities of work represented

by the quadrangles eadh and hdcg are work performed or

positive, those representedby gchf

andfhae

are work absorbed

or negative. The latter amount is larger than the former, and

the remainder, as represented by the quadrangle ahcd, is in

this case work absorbed.

In addition the variable body has drawn the quantity of

heat Qj from the body K^, and has given out to the body K^

the quantity of heat Q^=Q^+ Q- Of the two parts of which

Q^ consists, the one Q corresponds to the work absorbed, and

is generated from it, whilst the other Q^ has passed over as

heat from the body K^ to the body K^. Hence the result of

the cyclical process may here be described as follows: the

quantity of heat Q is generated out of work, and is given off

to the body K^, and the quantity of heat Q^ has passed over

from the colder body K^ to the hotter body K^.

§ 3. Cyclical process in the case of a body composedpartly of liquid, partly ofvapour.

In the foregoing sections, although in describing the

cyclical process we made no assumptions limiting the nature

of the variable body, yet the graphic representation of the

process was made to correspond to the case of a perfect gas.'

It is perhaps as well therefore to examine the cyclical process.



7i ox THE MECHANICAL THEORY OF HEAT,

over again in the case of a body of a different kind, in order to

see how its appearance may vary with the nature of the body

operated on. We will select for this examination a body

which has not all its molecules in one and the same state inall its parts, but consists partly of liquid, partly of vapour at

the maximum density.

Let us suppose a liquid contained in an expansible en-

velope, but only filling a part of it, and leaving the remainder

free for vapour having the maximum density corresponding

to the existing temperature T^. The combined volumes of

liquid and vapour are represented in Fig. 10 by the abscissa

oe, and the pressure of the

vapoiir by the ordinate ea.

Now suppose the envelope to

yield to the pressure and en-

large, while at the same time

the liquid and vapour are

connected with a body K^ of

constanttemperature

T^.

Asthe volume increases, more

liquid becomes vaporised, but

the heat consumed in the

a



SECOND MAIN PRINCIPLE. 7o

volume oe ; and let this compression take place-, partly in con-

nection with the body K^, of constant temperature T^, to which

all the heat produced by condensation of vapour passes over,

so that the temperature T^ remains unaltered : partly apartfrom this body, so that the temperature rises. Let it also be

arranged, that the first compression shall extend only so far

(to oh) as that the decrease of volume he then remaining maybe just sufficient to raise the temperature again from T^ to T^.

During this first compression the pressure remains unaltered,

at the value gc ; the external work thus absorbed is therefore

represented by the rectangle gcdh. During the last compres-

sion the pressure increases, and is represented by the

isentropic curve da, which must end exactly at the point a,

since with the original temperature T^ we must also have the

original pressure ea. The external work absorbed in this

last operation is represented by hdae.

At the end of the operation the liquid and vapour are

again in their original condition, and the cyclical process is

complete. The surplus of the positive ^bove the negativeexternal work, or the external work W which has been gained

on the whole in the course of the process, is represented as

before by the quadrangle abed. To this work must correspond

the absorption of an equivalent quantity of heat Q ; and if we

denote by Q^ the heat imparted during the expansion, and by

Qj the heat given out during the contraction, we may put

Qj = Q+^j, and the final result of the cyclical process is

again expressed by saying, that the quantity of heat Q is

converted into work, and the quantity Q.^ has passed over fi;om

the hotter body K^ to the colder ^j.

This cyclical process may also be carried out in the reverse

direction, and then the quantity of heat Q will be generated out

of work, and given off to the body K^, while the quantity Q^

will pass over from the colder body /C, to the hotter K^.

In a similar manner cyclical processes of this kind may

be carried out with other variable bodies, and graphically

represented by two isothermal and two isentropic lines ; in

which cases, while the form of the curves depends on the

nature of the body, the result of the process is always of the

same kind, viz. that one quantity of heat is converted into

work, or generated out of work, and that another quantity

passes over from a hotter to a colder body, or vice versa.




The question now arises. Whether the quantity of heat

converted into work, or generated out of work, stands in a

generally constant proportion to the quantity which passes over

from the hotter to the colder body, or vice versd; or whetherthe proportion existing between them varies according to the

nature of the variable body, which is the medium of the

transfer.

§ 4. Carnot's view as to the work performed during a

Cyclical Process.

Camot, who was the first to remark that in the produc-

tion of mechanical work heat passes from' a hotter into a

colder body, and that conversely in the consumption of

mechanical work heat can be brought from a colder into a

hotter body, and who also conceived the simple cyclical process

above described (which was first represented graphically by

Clapeyron), took a special view of his own as to the funda-

mental connection of these processes *-

In his time the doctrine was still generally prevalent thatheat was a special kind of matter, which might exist within

a body in greater or lesser quantity, and thereby occasion

differences of temperature. In accordance with this doctrine

it was supposed that heat might change the character of its

distribution, in passing from one body into another, and

further that it could exist in different conditions, which were

denominated respectively 'free' and 'latent'; but that the

whole quantity of heat existing in the universe could neither

be increased nor diminished, inasmuch as matter can neither

be created nor destroyed.

Camot shared these views, and accordingly treated it as

self-evident that the quantities of heat, which the variable

body in the course of the cyclical process receives from and

gives out to the surrounding space, are equal to each other,

and consequently cancel each other.

Helays this

downvery

distinctly in § 27 of his work, where he says :" we shall

assume that the quantities of heat absorbed and emitted in

these different transformations compensate each other exactly.

This fact has never been held in doubt ; admitted at first with-

out reflection, it has since been verified in many instances by

• Bejlexions lur la puissance motriee du feu. Paris, 1824.



" "' SECOND MAIN PfllNCIPLK 77

experiments with the calorimeter. To deny it would be to sub-

Vert the whole theory of heat, which rests on it as its basis."

Now since on this assumption the quantity of heat exist-

ing in the body was the same after the cyclical process as

before it, and yet a certain amount of work had been achieved,

Carnot sought to explain this latter fact from the circum-

stance of the heat falling from a higher to a lower tempera-

ture. He drew a comparison between this descending passage

of heat (which is especially striking in the steam-engine,

where the fire gives off heat to the boiler, and conversely the

cold water of the condenser absorbs heat) and the falling of

water froma higher to a lower level, by means of which amachine can be set in motion, and work done. Accordingly

in § 28, after making use of the expression ' fall of water,' he

applies the corresponding expression ' fall of caloric' to the

sinking of heat from a higher to a lower temperature.

Starting from these premises, he laid down the principle

that the quantity of work done must bear a certain constant

relation to the 'passage of heat,' i.e. the quantity of heat

passing over at the time, and to the temperature of the bodies

between which it passes ; and that this relation is indepen-

dent of the nature of the substance which serves as a

medium for the performance of work and passage of heat.

His proof of the necessary existence of this constant relation

rests on the principle " That it is impossible to create moving

force out of nothing," or in other words, " That perpetual

motion is an impossibility."

This mode of dealing with the question does not accord

with our present views, inasmuch as we rather assume that

in the production of work a corresponding quantity of heat

is consumed, and that in consequence the quantity of heat

given out to the surrounding space during the cyclical process

is less than that received from it. Now if for the production

of work heat is consumed, then, whether at the same time

with this consumption of heat there takes place the passageof another quantity of heat from a hotter to a colder body,

or not, at least there is no ground whatever for saying that

the work is created out of nothing. Accordingly not only

must the principle enunciated by Carnot receive some modifi-

cation, but a different basis of proof from that used by him

must be discovered.



78 ON THK 'MECHANICAL THEORY OF HEAT.

§ 5. New Fundamental Principle concerning Heat.

Various considerations as to the conditions and nature of

heat had led the author to the conviction that the tendency

of heat to pass from a warmer to a colder body, and thereby

equalize existing differences of temperature (as prominently

shewn in the phenomena of conduction and ordinary radia-

tion), was so intimately Tsound up with its whole constitution

that it must have a predominant influence under all conceiv-

able circumstances. He thereupon propounded the following

as a fundamental principle :" Heat cannot, of itself, pass from

a colder to a hotter body."

The words'

ofitself,'

here used forthe

sakeof brevity,

require, in order to be completely understood, a further ex-

planation, as given in various parts of the author's papers.

In the first place they express the fact that heat can never,

through conduction or radiation, accumulate itself in the

warmer body at the cost of the colder. This, which was

already known as respects direct radiation, must thus be

further extended to cases in which by refraction or reflection

the course of the ray is diverted and a concentration of rays

thereby produced. In the second place the principle must

be applicable to processes which are a combination of several

different steps, such as e.g. cyclical processes of the kind

described above. It is true that by such a process (as we

have seen by going through the original cycle in the reverse

direction) heat may be carried over from a colder into a

hotter body : our principle however declares that simul-

taneously with this passage of heat from a colder to a hotter

body there must either take place an opposite passage of heat

from a hotter to a colder body, or else some change or other

which has the special property that it is not reversible, except

under the condition that it occasions, whether directly or

indirectly, such an opposite passage of heat. This simul-

taneous passage of heat ia the opposite direction, or this

special change entaUing an. opposite passage of heat, is thento be treated as a compensation for the passage of heat from

the colder to the warmer body ; and if we apply this concep-

tion we may replace'îhe words " of itself" by " without com-

pensation," and then enunciate the principle as follows

"A passage of heat from a colder to a hotter body cannot

take place without compensation."



SECOND MAIN peincipm:. 79

This proposition, laid down as a Fundamental Principle

by the author, has met with much opposition ; but, having

repeatedly had occasion to defend it, he has always been able

to shew that the objections raised were due to the fact that

the phenomena, in which it was believed that an uncompen-sated passage of heat from a colder to a hotter body was to

be found, had not been correctly understood. To state these

objections and their answers at this place would interrupt too

seriously the course of the present treatise. In the discus-

sions which follow, the principle, which, as the author believes,

is acknowledged at present by most physicists as being correct,

will be simply used as a fundamentalprinciple ;

butthe

author proposes to retixm to it further on, and then to consider

more closely the points of discussion which have been raised

upon it.

§ 6. Proof that the relation between the quantity of heat

carried over, and that converted into work, is independent of

the nature of the matter which forms the medium of the

change.

Assuming the foregoing principle to be correct, it may be

proved that between the quantity of heat Q, which in a cyclical

process of the kind described above is transformed into work

(or, where the process is in the reverse order, generated by

work), and the quantity of heat Q^, which is transferred at the

same time from a hotter to a colder body (or vice versS,), there

exists a relation independent of the nature of the variable

body which acts as the medium of the transformation and

transfer ; and thus that, if several cyclical processes are per-

formed, with the same reservoirs of heat K^ and K^^, but with

different variable bodies, the ratio j- will be the same for

all. If we suppose the processes so arranged, according to

their magnitude, that the quantity of heat Q, which is trans-

formed into work, has in all of them a constant value, thenwe have only to consider the nmgnitude of the quantity of

heat Qj which is transferred, and the principle which is to be

proved takes the following form :" If ^here two different

variable bodies are used, the quantity of heat Q transformed

into work is the same, then the quantity of heat Q^, which

is transferred, will also be the same."




Let there, if possible, be two bodies G and C (e.g. the

perfect gas and the combined mass of liquid and vapour,

described above) for which the values of Q are equal, but

those of the transferred quantities of heat are different, and

let these different values he called Q^ and Q\ respectively

(y^ being the greater of the two. Now let us in the first

place subject the body (7 to a cyclical process, such that the

quantity of heat Q is transformed into work, and the quantity

Qjj is transferred from K^ to K^. Next let us subject C to a

cyclical process of the reverse description, so that the quantity

of heat Q is generated out of work, and the quantity Q'^ is

transferred from K^ to K^. Then the above two changes,from heat into work, and work into heat, will cancel each

other ; since we may suppose that when in the first process

the heat Q has been taken from the body K^ and transformed

into work, this same work is expended in the second process in

producing the heat Q, which is then returned to the same body

K^. In all other respects also the bodies will have returned,

at the end of the two operations, to their original condition,

with one exception only. The quantity of heat Q\, trans-

ferred from K^ to K^, has been assumed to be greater than

the quantity Q^ transferred from K^ to K^. Hence these two

do not cancel each other, but there remains at the end a

quantity of heat, represented by the difference Q\ — Q^, which

has passed over from K^ to K^. Hence a passage of heat will

have taken place from a colder to a warmer body without any

other compensating change. But this contradicts the funda-

mental principle. Hence the assumption that Q\ is greater

than Qj must be false.

Again, if we make the opposite assumption, that Q^ is

less than Q^, we may suppose the body C to undergo the

cyclical process in the first, and G in the reverse direction.

We then arrive similarly at the result that a quantity of heat

Q^ — Q\ has passed from the colder body K, to the hotter K^,

which is again contrary to the principle.Since then Q^ can be neither greater nor less than Q^ it

must be equal to Q^; which was to be proved.

We will now give to the result thus obtained the mathe-

matical form most convenient for bur subsequent reasoning.

Since the quotient ^ is independent of the nature of the



SECOND MAIN PRINCIPLE. 81

variable body, it can only depend on the temperature of the

two bodies JT^ and ^j, which act as heat reservoirs. The

same will of course be true of the sum

This last ratio, which is that between the whole heat received

and the heat transferred, we shall select for further considera-

tion ; and shall express the result obtained in this section as

follows :" the ratio -^ can only depend on the temperatures

Tj and T^" This leads to the equation

I^'^c^'.t;) (2),

in which <^ {T^T^ is some function of the two temperatures,

which is independent of the nature of the variable body.

§ 7. Determination of the Function<f>(T^T^.

The circumstance that the function given in equation (2)

is independent of the nature of the variable body, offers a

ready means of determining this function, since as soon as wehave found its form foi; any single body it is known for all

bodies whatsoever.

Of all classes of bodies the perfect gases are best adapted

for such a deterniination, since their laws are the most accu-

rately known. We will therefore consider the case of a per-

fect gas subjected to a cyclical process, similar to that graphi-cally expressed in Fig. 8, § 1 ; which figure may be here repro-

duced (Fig. 11), inasmuch as a perfect gas was there taken as

an example of the variable

body. In this process the

gas takes up a quantity of

heat Q during its expansion

ah, and gives out a quantity

of heat Qj during its com-pression cd. These quanti-

ties we shall calculate, and

l—f

g' then compare with each

other.

For this purpose we must

first turn our attention to the volumes represented by the

c. 6

Fig. 11.




aBscissse oe, oh, of, off, and denoted by v^, v^, F,, F,, in order

that we may ascesrtain the relation between them. Now the

volumes v^v^ (represented by oe, oh) form the limits of that

change of volume to which the isentropic curve ad refers,

and which may be considered at pleasure as an expansion or

a compression. Such a change of volume, during which the

gas neither takes in nor gives out any heat, has been treated

of in § 8 of the last chapter, in which we arrived at the fol-

lowing equation (43), p. 62

2; \v)'

c„where T and v are the temperature and volume at any point

in the curve. Substituting for these in the present case the

final values T„ and v,, we have2'

S.ft\!=© (=^

In exactly the same way we obtain for the change of volume

represented by the isentropic curve be (of which the initial

and final temperatures are also T^T^ ;

T /F\*^fW (*

Combining these two equations we obtain

T^= -, or —'=—? (o).

We must now turn to the change of volume represented

by the isothermal curve ab, which takes place at the constant

temperature T^, and between the limits of volume v^ and Fj.

The quantity of heat received or given off during such a

change of volume has been determined in § 8 of the last

chapter, and by the equation (41) there given, p. 61, wemay put in the present case

Q, = ET,log'^K (6).




Similarly for the change of volume represented by the iso-

thermal curve do, which takes place at temperature T^ between

the limits of volume v^ and V^, we have

Q, = i?T,logJ' (7).

From these two equations we obtain by division

^ =^ (8)

V VsmcebyfS) —' = —*.

The function occurring in equation (2) is now determined,

since to bring this equation into unison with the last equation

(8) we must have

<^(2;rj =-JJ

(9).

'a

We can now use in place of equation (2) the more deter-

minate equation (8), which may also be written as follows :

&-^^ = (10).

, The form of this equation may be yet further changed, byaffixing positive and negative signs to Q^, Q^. Hitherto these

have been treated as absolute quantities, and the distinction

that the one represents heat taken in, the other heat given

out, has been always expressed in words. Let us now for

convenience agre6 to speak of heat taken in only, and to

treat heat given out as a negative quantity of heat taken in.

If accordingly we say that the variable body has taken in

during the cyclical process the quantities of heat Qj and Q,,

we must here conceive Q^ as a negative quantity, i.e. the same

quantity which has hitherto been expressed by —Q^. Onthis supposition equation (10) becomes

% + %' = (11>,




§ 8, Cyclical processes of a more complicated character.

Hitherto we have confined ourselves to cyclical processes

in which the taking in of quantities of heat, positive or

negative, takes place at two temperatures only. Such pro-

cesses we shall in future call for brevity's sake Simple

Cyclical Processes. But it is now time to treat of cyclical

processes, in which the taking in of positive and negative

quantities of heat takes place at more than two tempera-

tures.

We may first consider a cyclical process with heat taken

in at three temperatures. This is represented graphically by

the figure abcdefa (Fig. 12), which, as in the former cases,

consists of isentropic and isothermal curves only. These

curves are again drawn, by way of example, in the form

which they would take in the case of a perfect gas, but this.

g ^-ê

Fig. 12.

is not essential. The curve ab represents an expansion at.

constant temperature T^; be an expansion without taking in

heat, during which the temperature falls from T, to T^; cd

an expansion at constant temperature T^; de an expansion

without taking in heat, during which the temperature falls

from jTj to Tg ; ef a, compression at constant temperature T^

and lastly fa a compression withoiit taking in heat, during

which the temperature rises from T^ to T^, and which brings

back the variable body to its exact original volume. In thp




expansions ab and cd the variable body takes in positive

quantities of beat Q, and Q,, and in the compression ef the

negative quantity of heat Q^. It now remains to find a rela-

tion between these three quantities.

For this purpose let us suppose the isentropic curve 6c

produced in the dotted line eg. The whole process is thereby

divided into two Simple Processes abgfa and cdegc. In the

first the body starts from the condition a and returns to the

same again. In the second we may suppose a body of the

same nature to start from the condition e, and to r6tum to

the same again. The negative quantity of heat Q^, which is

taken in during the compression ef, we may suppose dividedinto two parts q^ and g,', of which the first is taken in during

the compressiongf,

and the second during the compression

eg. We can now form the two equations, corresponding to

equation (11), which will hold for the two simple processes.

These equations are, for the process abgfa,

^+53 ='p

~fp '

and for the process cdego

-tj '3

Adding these equations we obtain

-'i 'i 'a

or, since 93+9a=Qa'

.(12).

In exactly the same way we may treat a process in which

heat is taken in at four temperatures, as represented by the

annexed figure abcdefgha. Fig. 13, which again consists solely

of isentropic and isothermal lines. The expansions ab and cd,

and the compressions ef and gh, take place at temperatures

Tj, Tjj, Tj, T^, and during these times the quantities of heat

Qi>Qi> Qi> Qi ^^® taken inrespectively ; the two former being



86 ON THE MECHANICAL THEOEY OF HEAT.

positive, and the two latter negative. Ptoduce the isentropic

curves 6c and fg in the dotted lines ci and gk respectively.

Then the whole process is subdivided into three Simple Pro-

cesses akghja, hUfk,and cdeic, which may be supposed to be

1^—ê

Fig. 13.

carried out with three exactly similar bodies. We may sup-

pose the quantity of heat Q^, taken in during the expansion ah,

to be divided into two parts q^ and q^, corresponding to ex-

pansions ak and kh ; and the negative quantity Q^, taken ia

during the compression ef, to be likewise divided into q^ and

q^, corresponding to compressions if and ci. Then we can

form the followiag equations for the three simple processes :

First, for akgha

Secondly, for Icbifk

Thirdly, for cdeic

2>- + «5 =

'a 'a




Adding, we obtaan

orrp ~ rp ^ rr "^ m> "^1 -^H -^S -^4

.(13).

In exactly the same way any other cyclical process, which

can be represented by a figure consisting solely of isentropic

and isothermal lines, and which has any given number of

temperatures at which heat is taken in, may be made to yield

an equation of the same form, viz.

or generally•^l

-^2 -^3 -^4

4-0. .(11).

§ 9. Cyclical Processes, in which taking m of Heat and

change of Temperatv/re take place simultameously.

We have lastly to consider such cyclical processes as are

represented by figures not consisting solely of isentropic and

isothermal lines, but altogether general in form.

The mode of treatment is as foUows. Let point a in

Fig. 14.

Fig. 14 represent any given condition of the variable body ;

let pq be an arc of the isothermal curve which passes through




a, rs an arc of the isentropic curve wliicli passes througt thfe

same point. Now let the body un4ergo a variation which is

expressed by a pressure-curve not coinciding with either of

the above, but taking some other course such as be or de.

Then we may consider.such a variation as made up of a very

grSat number dfvery small variations, in which we have

alternately change of temperature without taking in of heat,

and taking in of heat without chahge of temperature. This

series of successive variations wiU be represented by a dis-

continuous line, made up of alternate elements of isothermal

and isentropic curves, as drawn in Fig. 15, along the course of

Fig. 15.

be and de. The smaller the elements of which thedis-

continuous curve is made up, the more closely will it coincide

with the continuous line, and if these are indefinitely small

the coincidence wiU be indefinitely close. In this case it can

only make an indefinitely small difference, in relation to the

quantities of heat taken in and their temperatures, if wesubstitute for the variation reprpsented by the continuous line

the indefinitely large number of alternating variations, which

are represented by the discontinuous line.

We are now in a position to consider a complete cyclical

process, in which the taking in of heat is simultaneous with

changes of temperature, and which may be represented

graphically by curves of any form whatever, or merely by a

'single continuoTls and closed curve, such as is drawn in

Fig. 16. The area of this closed curve represents the ex-




ternal work consumed. Let it be divided into indefinitely

thin strips by means of adjacent isentropical curves, as shewnby the dotted lines in Fig, 16. Let us suppose these curvesjoined at the top and bottom by indefinitely small elements

Fig. 16.

of isothermal lines, which cut the given curve, so that

throughout its length we have a broken line, which is every-

where in indefinitely close coincidence with it. By the above

reasoning we may substitute for the process represented by

the continuous line the other process represented by the

broken line, without producing any perceptible alteration in

the quantities of heat taken in, or in their temperatures. Fur-ther, we may again substitute for the process represented by

the broken fine an indefinitely great number of Simple Pro-

cesses, which will be represented by the indefinitely small

quadrangular strips, made up each of two adjacent isentropic

curves, and two indefinitely smaU elements of isothermal

curves. If then for each one of these last processes we form

an equation similar to (11), in which the two quantities

of heat are indefinitely small, and can therefore be denoted bydifferentials of Q ; and if all these equations be finally added

together ; we shall then obtain an equation of the same form

as (14), but in which the sign of summation is replaced by the

sign of Integration, thus :

-f ^ ^^^^-I'-





(91)

CHAPTER IV.

THE SECOND MAIN PRINCIPLE UNDER ANOTHER FORM, OR

PRINCIPLE OF THE EQUIVALENCE OF TRANSFORMATIONS.

§ 1. On the two different hinds of Transformations.

In the last chapter it was shewn that in a Simple Cyclical

Process two variations in respect to heat take place, viz. that

a certain quantity of heat is converted into work (or generated

out of work),and another

quantity ofheat

passesfrom a

hotter into a colder body (or vice versS,). It was found fur-

ther that between the quantity of heat transformed into

work (or generated out of work) and the quantity of heat

transferred, there must be a definite relation, which is

independent of the nature of the variable body, and therefore

can only depend on the temperatures of the two bodies which

serve as reservoirs of heat.

For the former of these two variations we have already

employed the word "transformation," inasmuch as we said,

when work was expended and heat thereby produced, or

conversely when heat was expended and work thereby pro-

duced, that the one had been " transformed " into the other.

We may use the word " transformation " to express the second

variation also (which consists in the passage of heat from one

body into another, which may be colder or hotter than

the first), inasmuch as we may say that heat of one tem-

perature " transforms " itself into heat of another tempera-

ture.

On this principle we may describe the result of a simple

cyclical process in the following terms : Two transformations

are produced, a transformation from heat into work (or vice

versS,) and a transformation from heat of a higher tempera-




ture to heat of a lower (or vice versS.). The relation between

these two transformations is therefore that which is to be ex-

pressed by the second Main Principle.

Now, in the first place, as concerns the transformation ofheat at one temperature to heat at another, it is evident at

once that the two temperatures, between which the trans-

formation takes place, must come under consideration. But

the further question now arises, whether in the trans-

formation from work into heat, or from heat into work, the

temperature of the particular quantity of heat concerned

plays an essential part, or whether in this transformation the

particular temperature is matter of indifference.

If we seek to deduce the answer to this question from the

consideration of a Simple Cyclical Process, as described above,

we find that it is too limited for our purpose. For since in

this process there are only two bodies which act as heat,

reservoirs, it is tacitly assumed that the heat which is trans-

formed into work is derived from (or conversely the heat

generated out of work is taken in by) one or other of these

same two bodies, between which the transference of heat also

takes place. Hence a definite assumption is made from the

beginning as to the temperature of the heat transformed into

work (or conversely generated out of work), viz. that it

coincides with one of the two temperatures at which the

transference of heat takes place ; and this limitation prevents

us from learning what influence it would have on the relation

between the two transformations if the first-mentioned tem-perature were to alter, while the two latter remained un-

altered.

To ascertain this influence, we may revert to those

more complicated cyclical processes, which have also been

described in the last chapter, § 8, and to the equations

derived from them. But in order to give a clearer and simpler

view of the question it is better to consider a single process

specially chosen for this investigation, and by its help to

bring out the second Main Principle anew in an altered

form.

§ 2. On a Cyclical Process of special form.

Let us again take a variable body, whose condition is

completely . determined by its volume and pressure, so that



THE EQUIVALENCE OF TRANSFORMATIONS. 93

we can represent its variations graphically in the manneralready described. We will once more by way of exampleconstruct the figure in the form it assumes for a perfect gas,

but without making in the investigation itself any limiting

assumption whatever as to the nature of the body.Let the body be first taken in the condition defined by

the point a in Fig. 17, its volume being given by the abscissa

Fig. 17.

oh, and its pressure by the 'ordinate ha. Let T be the tem-

perature corresponding to these two quantities, and deter-

mined by them. We will now subject the body to the follow-

ing successivevariations

(1) The temperature T of the gas is change.d to T^

which we will suppose less than T. This may be done byenclosing the gas within a non-conducting envelope, so that

it can neither take in nor give out heat, and then allowing it

to expand. The decrease of pressure caused by the simul-

taneous increase of volume and faU of temperature will be

represented by the isentropic curve ah; so that, when the

temperature of the gas has reached T,, its volume andpressure have become ovand ih respectively.

(2) The variable body is placed in communication with

a body iTj of temperature T,, and then allowed to expand still

further, but so that all the heat lost in expansion is restored by

K^. With respect to the latter it is assumed that, on account of

its magnitude or from some other pause, its temperature is hot




perceptibly altered by this giving out of beat, and may tbere-

fore be taken as constant. Hence the variable body will also

preserve during its expansion the same constant temperature

r,, and its diminution of pressure will be represented by an

isothermal curve be. Let the quantity of heat thus given off

by Z, be called Q^.

(3) The variable body is disconnected from K^ and

allowed to expand still further, without being able either to

take in or give out heat, until its temperature has faUen from

T^ to fj. Let this diminution of pressure be represented by

the isentropic curve cd.

(4) The variable body is placed in communication witha body K^, of constant temperature T^, and is then com-

pressed, parting with all the heat generated by the com-

pression to jKj. This compression goes on until K^ has

received the same quantity of heat Q,, as was formerly

abstracted from K . In this case the pressure increases ac-

cording to the isothermal curve de.

(5) The variable body is disconnected from K^, and

compressed, without being able to take in or give out heat,

until its temperature has risen from T^ to its original value

T, the pressure increasing according to the isentropic curve-

ef. The volume on, to which the body is brought by this

process, is less than the original volume oh, since the pressure

to be overcome, and consequently the external work to be

transformed into heat, is less during the compression de

than during the expansion be; so that, in order to restore

the same quantity of heat Q,, the compression must be con-

tinued further than would have been necessary merely to

annul the expansion.

(6) The variable body is placed in communication with a

body-^ of constant temperature T, and allowed to expand to its

original volume oh, the heat lost in expansion being restored

from K. Let Q be the quantity of heat thus required.- If

the body attains the original volume oh at the original tem-perature T, then the pressure must also revert to its original

value, and the isothermal curve, which represents this last

expansion, will therefore terminate exactly in the point a.

The above six variations make up together a Cyclical Pro-

cess, since the variable body is finally restored exactly to its



THE EQUIVALENCE OF TRANSFOEMATIONS. 95

original condition. Of the three bodies, K, K^, K^, which in

the whole process only come under consideration in so far as

they serve as sources or reservoirs of heat, the two first have

at the end lost the quantities of heat Q, Q^ respectively,

whilst the last has gained the quantity of heat Q^ ; this maybe expressed by saying that Q^ has passed from K^ to K^,

while Q has disappeared altogether. This last quantity of

heat must, by the first fundamental principle, have been

transformed into external work. This gain of external work

is due to the fact that in this cyclical process the pressure

during expansion is greater than during compression, and

thereforethe

positivework greater than the negative ; its

amount is represented, as is easily seen, by the area of the

closed curve abode/a. If we call this work W, we have

by equation (5a) of Chapter I.

Q=W.

It is easily seen that the above Cyclical Process embraces

as a special case the process treated of at the commencement

of Chapter III., and represented in Fig. 8. For if we makethe special assumption that the temperature T of the body Kis equal to the temperature 2", of the body K^, we may then

dp away witLK altogether, and use K^ instead. The result

of the process will then be that one part of the heat given

out by the body iT^ has been transformed into work, and the

other part has been transferred to the body R^, just as was the

case in the process above mentioned.

The whole of this cyclical process may also be carried

out in the reverse order. The first step will then be to

connect the variable body with K, and to produce, instead

of the final expansion fa of the former case, an initial

compression af: and similarly the expansions fe and ed,

and the compressions dc, cb, and ba will be produced one

after another, under exactly the same circumstances as the

converse variations in the former case. It is obvious thatthe quantities of heat Q and Qj^ will now be taken in

by the bodies K and K^ respectively, and the quantity

of heat Qi will be given ovi by the body K^. At the same

time the negative work is now greater than the positive, so

that the area of the closed figure now represents a loss of

work. The result of the reversed process is therefore that




the quantity of heat Q^ has been transferred from K^ to K^^,

And that the quantity of heat Q has been generated out of

work and given to the body K.

§ 3. On Uquipalent Transformations.

In order to learn the mutual dependence of the two

simultaneous transformations above described, viz. the trans-

ference of Qi, and the conversion into work of Q, we shall

first assume that the temperatures of the three reservoirs of

heat remain the same, but that the cyclical processes, through

which the transformations are effected, are different. This

may be either because different variable bodies are subjectedto similar variations, or because the same body is subjected to'

any other cyclical process whatever, subject only to the con-

dition that the three bodies K, K^ and K^ are the only bodies

which receive or give out heat, and also that of the two

latter the one receives just as much as the other gives out.

These different processes may either be reversible, as in the

case considered, or non-reversible ; and the law which governs

the transformations will vary accordingly. However the

modification which the law undergoes for non-reversible pro-

cesses can be easily applied at a later period, and hence for

the present we will confine ourselves to the consideration of

reversible processes.

For all such it follows from the Principle laid d.own in the

last chapter (p. 78) that the quantity of heat Q,, transferred

from ifJ to K^, must stand in a constant relation to thequantity Q transformed into work. For let us suppose that

there were two such processes, in which, while Q was the

same in both, Q was different : then we might successively

execute that in which Q^ was the smaller in the direct order,

and the other in the reverse. In this case the quantity of

heat Q, which in the first process would have been trans-

formed into work, would in the second process be transformed

again into heat and given back to the body K; and in other

respects also everything would at the conclusion be restored

to its original condition, with this single exception that the

quantity of heat transferred from K^ to K^ in the second pro-

cess, would be greater than the quantity transferred from K^to K^ in the first process. Thus on the whole we have a

transfer of heat from the colder body K^ to the hotter K -,







THE EQUIVALENCE OF TRANSPOBMATIONS. 99

and K^, amd the quantity of heat Q^ which passes between

them, remain the same, but that for the body K of tempera-

ture T is substituted another body K' of temperature T':

and let us call the heat generated out of work in this case Q'.

Then, corresponding to the formier equation, we have the

following

Q' ^/{T') + Q,x F{T„ T,) = (3).

Adding (2) and (3) and substituting from (1) we obtain,

- Q xf{T) + Q' xf{T') = (4).

Now let us consider, as is clearly allowable, that these twosuccessive processes make up together a single process ; then

in this latter the two transferences of heat between K^ andK^ will cancel each other and disappear from the result ; wehave therefore only left the transformation into work of the

quantity of heat Q, given off by K, and the generation out

of work of the quantity of heat Q' taken in by K'. These

two transformations, which are of the same kind, can however

be so broken up and re-arranged as to appear in the light

of transformations of different kinds. For if we simply hold

fast to the main fact, that the one body K has lost the

quantity of heat Q, and the other K' gained the quantity Q",

then the heat equivalent to the smaller of these two quanti-

ties may be considered as having been transferred directly

from K to K', and it is only the difference between the two

which remains to be considered as a transformation of workinto heat or vice versS,. For example, let the temperature Tbe higher than T; then the transference of heat on the

above view is from a hotter to a colder body, and is therefore

.positive. Accordingly the other transformation must be

negative, i.e. a transformation from heat into work : whence it

follows that the quantity of heat Q given off by K is greater

than the quantity Q' taken in by K'. Thus if we divide Qinto its two component parts ^ and Q — Q', then the first of

these will have passed over from K to K', and the second is

the quantity of heat transformed into work.

On this view the two processes appear as combined into a

single process of the same kindj for the circumstance that

the heat transformed into work is not derived from a third

body, but from one or other of the same two bodies, between

7—2



100 ON THK MECHANICAL THEORY OP HEAT.

whicli the transference of heat takes place, makes no

essential difference in the result. The temperature of the

heat transformed into work is optional, and can therefore have

the same value as the temperature of one of the two bodies

in which case the third body is no longer required. Accord-

ingly for the two quantities of heat Q' and Q— Q there must

be an equation of the same form as (2), namely

-Q^g xf{T) + Q'xF {T, T) = 0.

Eliminating Q by means of equation (4) and then striking

out Q', we obtain

F{TT')=f{T)-f{T) (5).

As the temperatures T and T' are any whatever, the function

of two temperatures jP'(T'2"), which holds for the second kind

of transformation, is thus shewn to agree, with the function of

one temperature/ (2'), which holds for the first kind.

For the latter function we will for brevity use a simpler

symbol. For this it is convenient, for a reason which will beapparent later on, to express by the new symbol not the

function itself, but its reciprocal. We wiU therefore put

^=7^0^/(20 = ^ (6),

so, that T is now ihe unknown function of temperature which

enters into the Equivalence-value. If special values of this

function have to be written down, corresponding to tempera-

tures Tj, T^, etc.', or T', T' , etc., then this can be done by

simply using the indices or accents for t itself. Thus equa-

tion (5) wiE become

i^(r, 2") = -,--.T T

Hence the second Main Principle of the MechanicalTheory of Heat, which in this form may perhaps be called

the principle of the Equivalence of Transformations, can be

expressed in the following terms

"K we call two transformations which may cancel each other

without requiring any other permanent change to take place,

Equivalent Transformations, then the generation out of work



THE EQUIVALENCE OP TRANSFORMATIONS. 101

of the quantity of heat Q of temperature<^nas tne equivalence-

value — ; and the transference of the quantity of heat Q from

temperature T, to temperature T, has the Equivalence-value

<-^>in which t is a function of temperature independent of the

kind of process by which the transformation is accomplished."

§ 5. Combined value of all the transformations whichtake place in a single Cyclical Process.

If we write the last expression of the foregoing section

in the form , we see that the passage of the quantity

of heat Q from temperature T to T^ has the same equiva-

lence-value as a double transformation of the first kind,

viz. the transformation of the quantity Q from heat of tem-perature T, into work, and again out of work into heat of

temperature T^. The examination of the question how far

this external agreement has its actual foundation in the

nature of the process would here be out of place ; but in any

case we may, in the mathematical determination of the

Equivalence-Value, treat every transference of heat, in what-

ever way it may have taken place, as a combination of two

opposite transformations of the first kind.

By this rule it is easy for any Cyclical Process however

complicated, in which any number of transformations of both

kinds take place, to deduce the mathematical expression

which represents the combined value of all these trans-

formations. For this purpose, when a quantity of heat is

given off by a heat reservoir, we have no need first to

enquire what portion of itis

transformedinto work, and

what becomes of the remainder ; but may instead reckon

every quantity of heat given off by the hea,t reservoirs which

occur in the cyclical process as being wholly transformed into

work, and every quantity of heat taken in as being generated

out of work. Thus if we assume that the bodies K-^, K^, K^,

etc. of temperatures T^, T^, T,, etc. occur as heat reservoirs, and

if Q^, Q^, Q^, etc. are the quantities of heat given off during




the Cyclical Process (in wliicli we will now consider quanti-

ties of heat taken in as negative quantities of heat given

out*), then the combined value of all the transformations,

which we may call N, will be represented as follows

j^=_i^_Q._^_etc.,Ta ^2 T3

or using the sign of summation,

iV— 2^(7).

T

It is here supposed that the temperatures of the bodies

K^, K^, K^, etc. are constant,' or at least so nearly so that

their variations may be neglected. If however the tempera-

ture of any one of the bodies varies so much, either through

the taking in of the quantity of heat Q itself,. or through any

other cause, that this variation must be taken into account,

then we must, for every element of heat dQ which is taken

in, use the temperature which the body has at the moment

of its being taken in. This naturally leads to an integration.

If for thje sake of generality we assume this to hold for all

the bodies, th.en the foregoing equation takes the following

form

f''^(8),"—It

in which the integral is to be taken for all the quantities of

heat given off by the different bodies.

§ 6. Proof that in a reversible Cyclical Process the total

value of all the transformations must he equal to nothing.

If th-e Oyclical Process under consideration is reversible,

then,however

complicated it

maybe, it

can be provedthat the transformations which occur in it must cancel each

other, so that their algebraical sum is equal to nothing.

* This choice of positive and negative signs for the quantities of heat

agrees with that which we made in the last chapter, where we considered aquantity of heat taken in by the variable body as positive, and a quantity

given out by it as negative ; for a quantity given out by a heat reservoir is

taken in by the variable body, and vice versa.



THE EQUIVALENCE OF TRANSFORMATIONS. 103

For let us suppose tliat this is not the case, i.e. that this'

algebraical sum has some other value ; then let us imagine

the following process applied. Let all the transformations

which take place be divided into two parts, of which the first

has its algebraical sum equal to nothing, and the second is

made up of transformations all having the same sign. Let

the transformations of the first division be separated out into

pairs,each composed oftwo transformations of equal magnitude

but opposite signs. If all the heat reservoirs are of constant

temperature, so that in the Cyclical Process there is only a

finite number of definite temperatures, then the number of

pairs which have to be formed will be also finite; but should thetemperatures of the heat reservoirs vary continuously, so that

the number of temperatures is indefinitely great, and therefore

the quantities of heat given off and taken in must be dis-

tributed in indefinitely small elements, then the number of

pairs which have to be formed will be indefinitely large.

This however, by our principle,, makes no difference. The

two transformations of each pair are now capable of being done

backwards by one or two Cyclical Processes of the form

described in § 2.

Thus in the first place let the two given transformations

be of different kinds, e.g. let the quantity of heat Q of tem-

perature T be transformed into work, and the quantity of

heat Q^ be transferred from a body K^ of temperature T, to

a body K^ of temperature T^. The symbols Q and Q^ are

here supposed to represent the absolutevalues of the quanti-

ties. Let it be also assumed that the magnitudes of the two

quantities stand in such relation to each other that the follow-

ing equation, corresponding to equation (2), will hold, viz.

Then let us suppose the Cyclical Process to be performed in

the reverse order, whereby the quantity of heat Q, of tem-

perature T, is generated out of work, and another quantity

of heat is tra,nsferred from the body K^ to the body K^. Thi§

lattier quantity must then be exactly equal to the quantity Q^,

given in the above equation, and the given transformations

have thus been performed backwards.

Again let there be one transformation from work into



104 lON THE MECHANICAL THEORY OF HEAT.

heat and one from lieat into work, e.g. let the quantity of

heat Q of temperature T be generated out of work, and

the quantity of heat ^ of temperature 2" be transformed

intowork, and

let fbese two stand in such relation to each

other that we may put

T T

Then let us suppose in the first place that the same process as

last described has been performed, whereby the quantity of

heat Q of teinperature T has been transformed into work, and

another quantity Q^ has been transferred from a body K^ to

another body K^. Next let us suppose a second process per-

formed in the reverse direction, in which the last-named

quantity Q^ is transferred back again from K^ to K^, and a

quantity of heat of temperature T' is at the same time gene-

rated out of work. This transformation from work into heat

must, independently of sign, be equivalent to the former

trajisformation from heat into work, since they are both equi-

valent to one and the same transference of heat. The quantity

of heat of temperature T', generated out of work, must there-

fore be exactly as great as the quantity Q' found in the above

equation, and the given transformations have thus been madebackwards.

Finally, let there be two transferences of heat, e.g. the

quantity of heat Q transferred from a body K^ of tempera-

ture Tj to a body K^^ of temperature T^, and the quantity

Q\, from a body K\ of temperature T'j to a body K\ of tem-

perature T\, and let these be so related that we may put

«.(i-i).e'.(^-ij=o.

Then let us suppose two Cyclical Processes performed, in one

of which the quantity Q,^ is transferred from K^ to K^, and

the quantity Q of temperature T thereby generated out of

work, whilst in the second the same quantity Q is again

transformed into work, and thereby another quantity of heat

transferred from K\ to K\. This second quantity must then

be exactly equal to the given quantity Q\, and the twogiven transferences of heat have thus been done backwards.



THE EQUIVALENCE OP TRANSFORMATIONS. 105

When by operations of this kind all the transformations

of the first division have been done backwards, there then

rernain the transformations, all of the like sign, of the second

division, and no others whatever. Now first, if these trans-

formations are negative then they can only be transformations

from heat into work and transferences from a lower to a bigher

temperature; and of these the transformations of the *first

kind may be replaced by transformations of the second kind.

For if a quantity of heat Q of temperature T is transformed

into_ work, then we have only to perform in reverse order the

cyclical process described in § 2, in which the quantity of

heat Q of temperature T is generated out of work, and at thesame time another quantity $j is transferred from a body Kôf temperature T^ to another body K^ of the higher tempera-

ture Tj^. Thereby the given transfonnation from heat into

work is done backwards, and replaced by the transference of

heat from K^ to Ky By the application of this method, weshall at last have nothing left except transferences of heat

from a lower to a higher temperature which are not com-

pensated in any way. As this contradicts our fundamental

principle, the supposition that tlie transformations of the

second division are negative must be incorrect.

Secondly, if these transformations were positive, then

since the cyclical process under consideration is reversible,

the whole process might be performed in reverse order;

in which case all the transformations which occur in it

would take the opposite sign, and every transformation of

the second division would become negative. We are thus

brought back to the case already considered, which has been

found to contradict the fundamental principle.

As then the transformations of the second division can

neither be positive nor negative they cannot exist at all ; and

the first division, whose algebraical sum is zero, must em-

braceall

the transformations.

whichoccur in

the cyclicalprocess. We may therefore write iV= in equation (8), and

thereby we obtain as the analytical expression of the Second

Main Principle of the Mechanical Theory of Heat for reversi-

blg processes the equation

'"'^^= 0...,, ,

,(VII).'T




§ 7. On the Temperatures of the various quantities of

Heat, and the Entropy of the Body.

In the development of Equation VII. the temperatures of

the quantities of heat treated of were determined by those of

the heat reservoirs from which they came, or into which they

passed. But let us now consider a cyclical process, which is

such that a body passes through a series of changes of

condition and at last returns to its original state. This

variable body, if placed in connection with the heat reservoir

to receive or give off heat, must have the same temperature

asthe

reservoir ;

forit is only in this case that the

heat canpass as readily from the reservoir to the body as in the reverse

direction, and if the process is reversible it is requisite that

this should be the case. This condition cannot indeed be

exactly fulfilled, since between equal temperatures there can in

general be no passage of heat whatever ; but we may at least

assume it to be so nearly fulfilled that the small remaining

dififerences of temperature may be neglected.

In this case it is obviously the same thing whether weconsider the temperature of a quantity of heat which is being

transferred as being equal to that of the reservoir or of the

variable body, since these are practically the same. If how-

ever we choose the latter and suppose that in forming Equa-

tion VII. every element of heat Q is taken of that tem-

perature which the variable body possesses at the moment it

is taken in, then we can now ascribe to the heat reservoirs

any other temperatures we please, without thereby making

any alteration in the expression|

— . With this assumption

as to the temperatures we may consider Equation VII. as

holding, without troubling ourselves as to whence the heat

comes which the variable body takes in, or where that goes

which it gives off, provided the process is on the whole a re-

versible one.

The expression — , if it be understood in the sense lustT

given, is the differential of a quantity which depends on the

condition of the body, and at the same time is fully deter-

mined as soon as the condition of the body at the momentis known, without our needing to. know the path by which



THK EQUIVALENCE OF TRAKSFOllMATIONS. 107

the body has arrived at that condition ; for it is only in this case

that the integral will always become equal to zero as often as

the body after any given variations returns to its original con-

dition.

In another paper*, after introducing a further de-velopment.of the equivalence of transformations, the author

proposed to call this quantity, after the Greek word Tpoirrj,

Transformation, the Entropy of the body. The complete

explanation of this name and the proof that it correctly

expresses the conditions of the quantity under consideration .

can indeed only "be given at a later period, after the develop-

ment just mentioned has been treated of; but for the sake

of convenience we shall use the name henceforward.

If we denote the Entropy of the body by ;Si we may put

T

or otherwise

dQ = rdS., (VIII).

§ 8. On the Temperature Fimction t.

To determine the temperature function r we will apply the

same method as in Chaptei* III. § 7, p. 81, to determine the

function 4> (T^, T^. Eor, as the function t is independent of

the nature of the variable body used in the cyclical process, wemay, in order to determine its form, choose- any body we

please to be subjected to the process.

Wewill therefore

again choose a perfect gas, and, as in the above-mentioned

section, suppose a simple process performed, in which the gas

takes in heat only at one temperature T, and gives it out

only at another T^. The two quantities of heat which are

taken in and given out in this case, and whose absolute

values we may call Q and Q^, stand by equation (8) of the

last chapter, p. 83, in the following relation to each other

QrT,^^^-

On the other hand, if we apply Equation VII. to this

simple cyclical process, whilst at the same time we treat the

* Fogg. Ana^ Vol. cxxv. p. 390.




giving out of the quantity of heat Q as equivalent to the

taking in of the negative quantity — Q, we have the follow-

ing equation

^^»"^^. ™-

From equations (9) and (10) we obtain

or ^^^T (U).

If we now take T as being any temperature whatever and 2\

as some given temperature, we may write the last equation

thus

T=Tx Const (12),

and the temperature function t is thus reduced to a constant

factor.

What value we ascribe to the constant factor is indiffe-

rent, since it may be struck out of Equation VII. and thus

has no influence on any calculations performed by means of

the equation. We will therefore choose the simplest value,

viz. unity, and write the foregoing equation

r=T ,....(13).

The temperature function is now nothing more than the

absolute temperature itself.

Since the foregoing determination of the function t rests

on the equations deduced for the case of gases, one of the

foundations on which this determination rests will be the

approximate assumption made in the treatment of gases,

viz. that a perfect gas, if it expand at constant temperature,

absorbs only just so much heat as is required for the

external work thereby performed. Should anyone on this

account have any hesitation in regarding this determination

as perfectly satisfactory, he may in Equations VII. and VIII.

regard t as the symbol for the tempierature function as yet



THE EQUIVALENCK OF TEANSFOEMATIONS. , 109

undetennined, and use the equations in that form. Any such

hesitation would not, in the author's opinion, be justifiable,

and in what follows T will always be used in the place of t.

Equations VII. and VIII. wUl then be written in the following

forms, which have already been given under Equations V.

and VI. of the last chapter, viz.

dQ = TdS.



CHAPTER V.

FORMATION OF THE TWO FUNDAMENTAL EQUATION*

• § 1. Discussion of the Variables which determine the

Condition of the Body.

In the general treatment of the subject hitherto adopted

we have succeeded in expressing the two main principles of

the Mechanical Theory of Heat by two very simple equations

numbered III. and VI. (pp. 31 and 90).

dQ =dU+dW ..(Ill),

dQ = TdS (VI).

We will now throw these equations into altered forms

which make them more convenient for our further calcula-

tions.

Both equations relate to an indefinitely small alterationof condition in the body, and in the latter it is further

assumed that this alteration is affected in such a way as to

be reversible. For the truth of the first equation this assump-

tion is not necessary : we will howevpr make it, and in the

following calculation will assume, as hitherto, that we have

only to do with reversible variations.

We suppose the condition of the body under considera-

tion to be determined by the values of certain magnitudes,

and for the present we will assume that two such magnitudes

are sufficient. The cases which occur most frequently are

those in which the condition of the body is determined by its

temperature and volume, or by its temperature and pressure,

or lastly by its volume and pressure. We will not however

tie ourselves to any particular magnitudes, but will at first



FORMATION OF THE TWO FUNDAMENTAL EQUATIONS. Ill

assume that the condition of the body is determined by anytwo magnitudes which may be called x and y ; and these mag-nitudes we shall treat as the independent variables of our

calculations. In special cases we are of course always free to

take one or both of these yariables as representing either oneor two of the above-named magnitudes. Temperature, Volumeand Pressure.

If the magnitudes x and y determine the condition of

the body, we can in the above equations treat the Energy IT

and the Entropy S as being functions of the variables. Inthe same way the temperature T, whenever it does not itself

form one of these variables, may be considered as a functionof the two variables. The magnitudes W and Q on the con-

trary, as remarked above, cannot be determined so simply,

but must be treated in another fashion.

The differential coefficients of these magnitudes we shall

denote as follows

dW dW

f = ^= f =^ (^)-

These differential coefficients are definite functions of xand y. For suppose the variable x is changed into x+dxwhile

yremains constant, and that this alteration of condi-

tion in the body is such as to be reversible, then we are

dealing with a completely determinate process, and the

external work done in that process must therefore be also

dWdeterminate, whence it follows that the quotient -^ must

equally have a determinate value. The same will hold if wesuppose y to change to 1/ + dy while x remains constant. If

then the differential coefficients of the external work W are

determinate functions of x and y it follows from Equation III.

that the differential coefficients of the quantity of heat Qtaken in by the body are also determinate functions of x

and y.

Let us now write for dW and dQ their expressions as

functions of dx and dy, neglecting those terms which are of a




higher order than doa and dy. We then have,

dW= mdx + ndi/ (3),

dQ = Mdx + Ndi/ (4),

"

and we thus obtain two complete differential equations, which

cannot be integrated so long as the variables a; and y are

independent of each other, since the magnitudes m, n and

M, N do not fulfil the conditions of integrability, viz.

dm _dn,

dM_dNdy

~ dx dy dx'

The magnitudes W and Q thus belong to that class which

was described in the mathematical introduction, of which the

peculiarity is that, although their differential coefficients are

determinate functions of the two independent variables, yet

they themselves cannot be expressed as such functions, and

can only be determined when a further relation between the

variables is given, and thereby the way in which the varia-

tions took place is known.

§ 2. Elimination of the quantities U and S from the

two Ftmdamental Equations.

Let us now return to Equation III., and substitute in it

for dW and dQ expressions (3) and (4) ; then, collecting to-

gether the terms in dx and dy, the equation becomes,

Mdx + Ndy = (^+ m^ dx+(~ + «) dy.

As these equations must hold for all values of dx and dy,

we must have,

iV=-5-H-W.



FORMATION OF THE TWO FUNDAMENTAL EQUATIONS. 113

Differentiating the first equation according to y, and the

second according to x, we obtain,

dM^ffU_ dm

dy dxdy dy

dN^ cP£ dn

dx dydx dx

We may apply to JJ the principle which holds for every

function of two independent variables, viz. that if they are

differentiated according to both variables, the order of dif-

ferentiation is a matter of indifference. Hence we have

d'u ^ dru_

dxdy dydx

Subtracting one of the two above equations from the

other we 'obtain,

dM dN_ dm_dndy dx dy dx

^

We may now treat Equation VII. [in the same manner.

Putting for dQ and d8 their complete expressions, it be-

comes,

Mdx + Ndy = T ij- dx -^ -^ dyj ,

M^ N. dS, dS,or

Y'^^'^T^^'^^'^'^d^'^'''-

This equation divides itself, like the last, into two, viz.

M^dST dx'

F_d8T dy'

Differentiating the first of these according to y, and the

second according to x, we obtain,



114' ON THE MECHANICAL THEORY 01? HEAT.

T^ _dxdy

dx dx _ d'S

: T^ dydx'

d'S d'S

But as before,

dxdy dydx

hence subtracting one of tbe above equations from the other

we obtain

dy dy dx dx _^

dM dN Ifji^dT ,rdT\ ,.

l^-d^ = T[^dy-^dx)^^^-

The two equations (5) and (6) may be, also written in a

somewhat different form. To avoid the use of .so many-

letters ill the formula, we will replace M and N, which were

introduced as abbreviations for -^ and -i^, bythose differen-

tial coefficients themselves. Similarly for m and n we will

dW ' dWwrite -i— and t— , Then the right hand side of equation,

dx dy°

(5) may be wiitten

d^ /dW\ _ d^(dW\

dy\dx J dx\dy J

'

Thus the magnitude represented by this expression is a func-

tion of X and y, which may generally.be considered as known,

inasmuch as the external forces acting on the body are open

to direct observation, and from these the external work can

be determined. The above difference, which will occur very-

frequently from henceforward, we shall call "The Work




Difference referred to x and y" and shall use for it a special

symbol, putting

J. _± (dW\ _ d_ (dW\

. .. ^~dy\dx) dxKdyj^^^•

Through these changes equations (5) and (6) are trans-

formed into the following

A. f^\ _ i. f^\ - n rfi\

dy\dx) dx\dyj~ '"^>-'

^(dQ\_±fdQ\^lfdT^dQ_dT^dQ\

dy\dx) dx\dyj T\dy dx dx dyj" ''

These two equations form the analytical expressions of

the two fundamental principles for reversible variations, in

the case in which the condition of the body is determined by

two given variables. From these equations follows a third,

which is so far simpler as it contains only differential coeffi-

cients of Q of the first degree, viz.

^Jx|f-Sx|=-..- (-)•

§ 3. Case in which the Temperature is taken as one of the

Independent Variables.

The above three equations are very much simplified if the

temperature of the body is selected as one of the independent

variables. Let us for this purpose put y = T, so that the twoindependent variables are the temperature T, and the still

undetermined quantity x. Then we have first,

^=1dy

dTAgain, referring to the differential coefficient ;,- , it is

assumed in its formation that, while x is changed into x + dx,

the other variable, hitherto called y, remains constant. But

as T is now itself the other variable, it follows that T must

remain constant in the above differential coefficient, or in

other words

dx

8—2



116 ON THE MECHANICAL THEORY OP HEAT.

Again, if we form the Work-difference referred to x and

T, this wUl run as follows

ê^m\_d_idw\^"^ dT\dx) dxKdTj ^ '

Hence equations (8), (9), (10) take the following form;

^©-s(S)-^« ('^)-

±(dQ\_d_(d^\^ldQ „,

dT\dx) dxKdT) Tdx ^ ''

S=^^- (1^)'

If the product TD^^,, given in equation (14), be substi-

tuted for -j^ in equation(12),

and then differentiated accord-

ing to T, the following very simple equation will be the

result

d fdQ\ dD,^

§ 4. Particular Assumptions as to the External Forces.

We have hitherto made no particular assumptions as tothe external forces to which the body is subjected, and to

which the external work done during its alterations of condi-

tion is related. We will now consider more closely a special

case, which occurs frequently in practice, namely' that in

which the only external force which exists, or at least the

only one which is of sufficient importance to be taken into

consideration, is a pressure acting on the surface of the body,

everywhere normal to that surface, and of uniform intensity.

In this case external work can only be done during changes

in the volume of the body. Let p be the pressure per unit

of surface, v the volume ; then the external work done,

where this volume is increased to ?; + dv, is given by the

equation

dW=pdv (16).



FOEMATION OF THE TWO FUNDAMENTAI, EQUATIONS. 117

Let US now suppose tlie condition of the body to be given

by the values of two given variables x and y. Then the

pressure p and volume v must be considered as functions of

X and y. We may thus write the last equation as follows

"^'^(S-^+l*)-





FORMATlbN- OF THE TWO FUNDAMENTAL EQUATIONS. 119

If p and T are taken :

d (dg\ d_ /dQ\ ^_dv^dT [dp] dp \dTj dT

d dQ_± (dQ\_l dQdT dp dp [dTJ

~ T dp'

dp dT'

dp [dTj dT'

.(26).

(27).

If V and p are taken

dp \dvj dv \dp J'

d_fdQ\_d/dQ\l^rdT dQ _dT dQ\

dp \dv J dv \dpj~ T \dp dv dv dp J

'

dp dv dv dp

§ 6. Equations in the case of a body which undergoes

a Partial Change in its Condition of Aggregation.

A case whict permits a still further simplification peculiar

to itself, and which is of special interest from its frequent

occurrence, is that in which the changes of condition in the

body are connected with a partial change in its Condition ofAggregation.

Suppo.se a body to be given, of which one part is in

one particular state of aggregation, and the remainder in

another. As an example we may conceive one part of the

body to be in the condition of liquid and the remainder

in the condition of vapour, and vapour of the particular

density which it assumes when in contact with liquid : the

equations deduced will however hold equally if one part

of the body is in the solid condition and the other in the

liquid, or one part in the solid and the other in the vaporous

condition. We shall, for the sake of generality, not attempt

to define more nearly the two conditions of aggregation

which we are treating of, but merely call them the first and

the second conditions.




Let a certain quantity of this, substance be inclosed in

a vessel of given volume, and let one part have the first,

and the other the second condition of aggregation. If the

specific volumes (or volumes per unit of weight), which the

substance assumes at a given temperature in the two dif-

ferent conditions of aggregation, are different, then in a

given space the two parts which are in the different con-

ditions of aggregation cannot be any we please, but can

only have determinate values. For if the part which

exists in the condition of greater specific volume increases

in size, then the pressure is thereby increased which the

inclosed substance exerts on the inclosing walls, and con-sequently the reaction which those walls exert upon it,

and finally a point will be reached, where this pressure is

so great that it prevents any further passage of the substance

into this condition of aggregation. When this point is

reached, then, so long as tHe temperature of the mass and

its volume, i.e. the content of the vessel, remain constant,

the magnitude of the parts which are in the two conditions

of aggregation can undergo no further change. If, however,

whilst the temperature remains constant, the content of

the vessel increases, then the part which is in the conditionj

of aggregation of greater specific volume can again increase

at the cost of the other, but only until the same pressure as

before is attained and any further passage from one condition

to the other thereby prevented.

Hencearises the peculiarity, which distinguishes

thiscase from all others. For suppose we choose the temperature

and the volume of the mass as the two independent variables

which are to determine its condition; then the pressure

is not a function of both these variables, but of the tempera-

ture alone. The same holds, if instead of the volume we

take as the second independent variable some other quantity

which can vary independently of the temperature, and which

in conjunction with the temperature determines fuUy thecondition of the body. The pressure must then be inde-

pendent of this latter variable. The two quantities, tem-

perature and pressure, cannot in this case be chosen as

the two variables which are to serve for the determination

of the body's condition.

We will now take, in addition to the temperature T, any



into the following:

dr'


other magnitude x, as yet left undetermined, for the second

independent variable which is to determine the body's

condition. Let us then consider the expression given in

equation (19) for the work-diflference referred to xT, viz.

T\ _dp dv dp . dv

''''~dT''d^~drx^dT-

In this we must, by what has been said, put -^ = 0,

and we thus obtain

^.-J4: (^«)-

The three equations (12), (13), (14) are thereby changed

'g:

d_ fdQ\ _ d^ fdQ\ _dp dv . .

iT\dool dx\dT)~ df dx ^ ''

ji/^N_^/rf<3\_l dQ

dT\dxj dx\dTj T dx ^ ''

S =^*x| (^'>-

§ 7. Clapeyron's Equation and Carnot's Function.

To conclude the developments of the Fundamental Equa-

tions which have formed the subject of the present chapter,

we may consider the equation which Clapeyron* deduced as

a fundamentajl equation from the theory of Carnot, in order

to see in what relation it stands to the equations we have'

here developed. As however Clapeyron's equation contains

an unknown function of temperature, usually designated as

Carnot's function, it will be advisable beforehand to throwour equations, so far as they will come under considera-

tion, into the form which they take, if the temperature

function t, introduced in the last chapter, is treated as still

indeterminate, and is not, in accordance with the value

* Journal de VEcole Folytechnique, Vol. xiy. (1834).




there determined, put equal to the absolute temperature T.

We shall thereby obtain the advantage of fixing the relation

between our temperature-function t and Carnot's Function.

If instead of equation

dQ=Td8,

we use the less determinate Equation VIII. of the last chapter,

dQ = rdS,

and eliminate 8 from this eque-tion, in the same manner

asin

§ 2, weobtain instead of

equation (9)the following :

£(dQ\_d^(dQ\l^dj^^dQ_dT^dQ\_

dy\dx) dx\dy) T\dy dx dx dyj" ''

Combining this with (8) we obtain instead of equa-

tion (16),

^^S-S"!"^- (»^>

If we now assume that the only external force is a

uniform and normal pressure on the surface, we may use

for D^y the expression given in equation (18), and the above

equation becomes

dr dO dr dQ {dip dv dp dv

— X -7^ — -, - X-^=T{-f x^ f X i) w-y dx dx dy \dy dx dx dy.

If further we choose as independent variables v and p,

putting x = v, and y = p, we obtain

— ^ _ !^ '^^ — r^ "I

dp dv dv dp ^ ''

But as T is only a function of T, we may put

dr^dr^ dT .dT_dT dT

dv dT^ dv dp~ dT^ dp-

Introducing these values of t- and -j- in the above equation,



FOEMATION OF THE" TWO FUNDAMENTAL EQUATIONS. 123

and dividing by ^ , we obtain, instead of the last of equa-

tions (27), the following:

dp dv dv dp dT^"' ''

dT

It is here assumed that the heat is measured in mechani-

cal units. To introduce the ordinary measure of heat, wemust divide the expression on the right-hand side by the

mechanical equivalent of heat, which gives:

dT^^dQ dT^JQ r

dp dv dv dpzpdr ^ ''

dl'

Clapeyron's equation agrees in form with this, since it is

written*

fx^_fx^=(7 (38),dp dv dv dp ^

where C is an undetermined function of temperature, and is

the same as Carnot's Function already mentioned.

If we equate the right-hand expressions of (.37) and (38),

we obtain the relation between C and t, as follows

^==-fd^=Zd^7[ ^^^^'

^dT ^—df~

If, adhering to the determination of t given by the author,

we assume it to be nothing more than the absolute tempera-

ture T, then C takes the simpler form,

c=| m.

As equation (33) is formedby the combination of two equa-

tions, which express the first and second Main Principles, it

* Pogg. Ann. Vol. lix. p. 574.



124 ON THE MECHANICAL THEORY OF HEAT. •

follows that Clapeyron's equation must be considered not as an

expression of the second Main Principle in the form assumed

by the author, but as the expression of a principle, which

may bededuced from the combination of the first and second

principles. As concerns the manner in which Clapeyron has

treated his differential equation, this differs widely from the

method adopted by the author. Like Carnot, he started from

the assumption that the quantity of heat which must be

imparted to a body during its passage from one condition to

another, may be fully ascertained from its initial and final

conditions, without its being necessary to know in what way

and by what path the passage has taken place. Accordinglyhe considered ^ as a function of p and v, and deduced by

integrating his differential equation the following expression

for Q'.

Q=F{T)-C,t>{p,v) (41),

in which F{T) is any function whatever of the temperature

and ^(p,v) is a, function of p and v which satisfies the follow-

ing more simple differential equation :

f:.#_f+^/=i (42).av dp dp av

^

To integrate this last equation we must be able to express the

temperature T for the body in question as a function of p

andV.

If we assumethat the

body in question is a perfectgas, we have

2^=5 (43).

whence

dT_p dT_v_

dv B dp B'

Hence equation (42) becomes

^^-^f=^ (**)•

Integrating, we have

^ {p> v) = Rlogp + ^ ipv),




where ^(pv) is any function whatever of the product pv. For

this we may by equation (43) substitute any function what-

ever of the temperature, so that the equation becomes,

(^{pv) = Blogp + ^fr{T) (45).

If we introduce this expression for <p {p, v) in (41), and put

where B again expresses any function whatever of the tem-

perature, we obtain,

Q = B{B-Clogp) (46).

This is the equation which Clapeyron has deduced for the

case of gases.



CHAPTER VI,

APPLICATION OF THE MECHANICAL THEORY OF HEAT TO .

SATURATED VAPOUR.

§ 1. Fundamental equations for saturated vapour.

Among the equations of the last chapter, those deduced

in § 6, which refer to a partial change in the body's state of

aggregation, may conveniently be treated first; inasmuch as

the circumstance there mentioned, viz. that the pressure is

only a function of the temperature, greatly facilitates . the

treatment of the subject. We will in the first place consider

the passage from the liquid to the vaporous condition.

Let a weight M of any given substance be inclosed in an

expansible envelope: of this let the partm be in thecondition

of vapour, and that vapour (as necessarily follows from its

contact with the liquid) at its maximum density; and let the

remainder M— m be liquid. If the temperature T of the

mass is given, the condition of the vaporous part, and at thesame time that of the liquid part, is thereby determined. If

m be also given and thereby the magnitudes of both parts

known, then we know the condition of the whole mass. Wewill accordingly choose T and m as the independent variables,

and will substitute m for x in equations (29), (30), (.31) of

the last chapter. Then these equations become

dT\dm) dm [dTJ~ dT^ dm W'

d^fdQ\ _ d_ fdQ\ _l^dQdT\dm) dm\dTj T dm ^'^^'

dQ^j^dp dv^

dm dT dm ^"^J-



SATUEATED VAPOUR. 127

We may now denote the specific volume (i.e. the volumeof a unit of weight) of the saturated vapour by s, and the

specific volume of the liquid by a: Both these magnitudes

bear some relation to the temperature T and its correspond-ing pressure, and are therefore, like the pressure, functions of

the temperature alone. If we further denote by v the total

volume of.the mass, we may then put

v='nis+ {M—m) a-,

= m{s — a) + Mcr.

We will substitute for the difference (s — cr), a simpler ex-

pression, by putting

u = s-a- (4),

whence it follows that

V = mu + Ma- (5)

whence

^^

;^'=" "(6)'

The quantity of heat which must be applied to the mass, if

a unit of weight of the substance, at temperature T and

under the corresponding pressure, is to pass from the liquid

into the vaporous condition, and which may be shortly called

the vaporizing heat, may be denoted by p ; then we have

dQ ,„dm = P (')•

We will further introduce into the equations the specific heat

of the substance in the liquid and vaporous condition. Thespecific heat here treated of is not however that at constant

volume, nor yet that at constant pressure, but belongs to the

case in which the pressure increases with the temperature in

the same manner as the maximum expansive power of the.saturated vapour. This increase of pressure has very little

influence on the specific heat of the liquid, since liquids are

but slightly compressible by such pressures as are herein con-

sidered. We shall hereafter explain how this influence maybe calculated, in our researches on the different kinds of

specific heat, and a single example will suffice here. For

water at boiling-point the difference between the specific




heat here considered and the specific heat at constant

pressure, is only of the latter, a difference which may be

neglected. Accordingly, we may for the purposes of calcula-

tion take the specific heat of the liquid here considered as

being equal to the specific heat at constant pressure, although

their meaning is different. We will call this specific

heat 0.

With vapour it is otherwise. The specific heat here con-

sidered refers, as shewn above, to that quantity of heat which

saturated vapour requires to heat it through 1", if it is at the

same time so powerfully compressed that even at the higher

temperature it again returns to the saturated condition. Asthis compression is very considerable, this kind of specific

heat is very different from all which we have hitherto treated

of We shall call it the Specific Heat of Saturated Steam,

and shall denote it by H.

Bringing in the two symbols G and H, we may now at once

"write down the quantity of heat which is necessary to givethe increase of temperature dT to the quantity of vapour

m, and the quantity of liquid M— m. The result will be as

follows

mHdT+(M-m)CdT,

whence

or otherwise

'^ =mH+{M-m)C,

^= m{H-G)+MG (8).

From equations (7) and (8) we have

d^(dQ\ dp

U^h^-' (^»>-

Substituting in equations (1, 2, 3) the values given in equa-tions (7, 9, 10) we have



SATUBATED VAPOUR. 129

%^C-H=u% (U).

^+a-F=|(12),

P = Tu^T ^^^)-

These are the fundamental equations of the Mechanical

Theory of Heat as relates to the generation of vapour. Equa-

tion (11) is a deduction from the first fundamental principle,

(12) from the second, and (13) from both together.

If it is desired to use the ordinary and not the mechanical

measures for the quantities of heat, we need only divide all

the members of the foregoing equations by the mechanical

equivalent of heat. In this case we will denote the two

specific heats and the hjeat of vaporization by new symbols,

putting

c = ^; ^ = 2;! '' =A'

(^*)-

The equations then become

dr , u fdp\ ,, .,

dT^'-^ÊKdT)^^'^'

_+C-A.= y (16),

§ 2. Specific Heat of Saturated Steam.

As the foregoing equations (15), (16) and (17), of which

however only two are independentjhave thus been obtained by

means of the Mechanical Theory of Heat, we may make use.of

them in order to determine more closely two magnitudes, of

which one was previously quite unknown and the otber only

known imperfectly; viz. the magnitude h and the magnitude

s contained in m.

If we first apply ourselves to the magnitude h, or the

Specific Heat of Saturated Steam, it may be worth while in

c. 9



ISO ON THE MECHANICAL THEORY OF HEAT.

the first place to give some account of the views formerly

promulgated concerning this magnitude.

The magnitude h is of special importance in the theory

of the steam engine, and in fact the first who published, any

distinct views upon it was the celebrated improver of the

steam engine, James Watt. In his treatment of the subject

he naturally started from those views which were based on the

older theory of heat. To this class belongs especially the idea

mentioned in Chapter I., viz. that the so-called total Jheat, i.e.

the total quantity of heat taken in by a body during its passage

from a given initial condition to its present condition, depends

only on the present condition and not on the way in whichthe body has been brought into it ; and that it accordingly

may be expressed as a function of those variables on which

the condition of the body depends. According to this view

we must in our case, in which the condition of the body com-

posed of liquid and vapour is determined by the quantities

T and m, consider this quantity of heat ^ as a function of Tand m ; accordingly we have the equation

A (^ _A (i^ = ndT \<hnj dm [dTj

"'

If we here substitute for the two second differentials their

values given in equations (9) and (10), we have

dp

or dividing by JE

^T+ C-B=0,

— +c-h = 0,

whence we have, to determine h, the equation,

^ =^ + c (18). .

This was in fact the equation which was formerly used to

determine h, though not quite in the same form. To calculate

fi from this equation we must know the differential co-

efficient ^, or the change of the vaporizing heat for a given

change of temperature,



SATURATED VAPOUR. 131

Watt had made experiments on the vaporizing heat of

water at different temperatures, and was thereby led to a

result, which may be expressed by a very simple law, com-

monly called Watt's law. This in its shortest form is as

follows :" The sum of the free and ktent heat is always

constant." By this is meant that the sum of the two quan-

tities of heat, which must be imparted to a unit of weight of

water, in order to raise it from freezing point to temperature

T, and then at that temperature convert it into steam, is in-

dependent of the temperature T itself. The quantity of

heat required for heating the water is expressed by the

integral

in which a is the absolute temperature of freezing point.

The heat of vaporization is represented by r. Watt's law

therefore leads to the equation

r-l-f C£Z!Z'= Constant (19).

Differentiating,

J+c = (20),

, combining this with equation (18) we have

fe = (21).

This result was long considered as correct, and was expressed

by the' following principle : If steam of maximum density

changes its volume within a vessel impermeable to heat, it

always preserves its maximum density.

More recentlyEegnault made very careful experiments on

the changes in the heat of vaporization with changes in the

temperature* ; and thereby discovered that Watt's law, accord-

ing to which the sum of the free and latent heat is always

constant, does not agree with the facts, but that this sum has-

an increasing value as the temperature rises. The result of

his experiments is expressed in the following equation, in

* Relations de Experiences, t. i. ; Mem. de VAcad., t. ixi., 1847.

9-2




which instead of the absolute temperature T we introduce

the temperature t reckoned from the freezing point

•+ [ccif= 606-5 + 0-305 i (22).•'

Differentiating this equation with regard to t, and then sub-

stituting-J

for -j- , both having the same value, we have

^+c = 0-305 (2.3).

Combining this equation with (18) we have

^=-305 (24).

This 'was the value of h, which it was supposed, after the

publication of Eegnault's experiments, must be substituted

for the value zero and applied to the theory of the steam

engine. Hence arose the idea that if saturated steam be

compressed, and thereby heated, in such a way that it always

has exactly the temperature for which the density is a maxi-mum, it must take in heat from without ; and conversely in

expanding, in order to cool itself in a manner corresponding

to the expansion, it must give out heat from itself Hencewe must further conclude that if saturated steam be com-

pressed in a vessel impermeable to heat, a fall of temperature

must take place ; whilst in expanding the steam will not' re-

main at its maximum density, inasmuch as its temperature

cannot fall so low as this will necessitate.

Having thus explained the conclusions previously drawn

in relation to h, we will now see what may be concluded

from the equations here developed.

The quantity h occurs in the two equations (15) and (16);

but the first of these also contains the quantity u, which cannot

at present be considered as accurately known ; it is therefore

lessconvenient for determining h than the latter, which in

addition to A contains only such quantities as the experi-

ments of Regnault have determined with great accuracy in

the case of water, and of many other fluids. This equation

may be written

, dr r

^^dT'^''~T• f^25)-




We have thus obtained by the Mechanical Theory of Heat a

new equation for determining h, which differs from the equa-

tion (18), previously assumed to be correct, by the negative

Tquantity—^, the value of which quantity is thus of great

importance.

§ 3. Numerical Value of h. for Steam.

K we applj' equation (25) to the case of water, we must first,

following Regnault, give to the sum of the first two symbols

on the right-hand side the value 0'305. To determine the

last symbol we must know the value of r, as a function of the

temperature. Equation (22) gives us

»• = 606-5 -t-0-305«-fcd{ (26).Jo

The specific heat of water c is determined according to

Regnault by the following formulae :

c = 1 -f- 0-00004!! + 00000009«'' (27).

Applying this, equation (26) becomes

r= 606-5 - 0-695f- 0-0000«''-0-0000003f...(28).

Substituting this value for r in (25), and replacing T by

273

+1, we obtain for steam the following equation :

i no«- 606-5 -0-695«-000002f- 0-0000003*' ,„„,A = 0-30o 2^3-j-^ ...(29).

The expression for r given in (28) is inconvenient from its

length, and the experiments on the heat of vaporization at

different temperatures, valuable as they are, can scarcely

possess such a degree of accuracy as to make so long a

formula necessary to represent them. Accordingly in his

paper on the theory of the steam engine the author pre-

ferred to use the following formula

r = 607-0-708< (30).

The manner in which the two constants in the formula,

are determined will be more closely examined further on,




in describing the steam engine. Here "vve will only give a

comparison of some values determined by both formulse, in

order to shew that the difference between them is so small,-

that one may be substituted for the other without danger

*



SATURATED VAPOUR, 135

only that it remains exactly in the condition of saturated

steam. Accordingly if this last condition is to hold heat mustbe imparted during the expansion.

Should the original saturatedsteam be

contained inavessel impermeable to heat, it will be superheated during

compression and will be in part condensed during expan-

sion.

The conclusion that the specific heat of saturated steam

is negative was drawn by Rankine and by the author inde-

pendently, and about the same time*. Eankine however

developed only the first of the two equations (15) and (16),

which contain A, and this in a somewhat different form.

The second it was impossible for him to develope, since he was

without the second fundamental principle, which was neces-

sary thereto. Since in the first equation there occurs together

with h the specific volume of the saturated steam, which is

contained in u, Rankine in order to determine this applied

to saturated steam the law of Mariotte and Gay-Lussac,

which, as we shall see further on, is inaccurate. More exact

determinations of h could only be accomplished by means of

equation (16), which was first established by the author.

§ 4. Nwmerical Vcdue of h for other Vapours.

When equation (25) was first published, Eegnault's deter-

minations of the specific heat and heat of vaporization as func-

tions of temperature had been performed only for the case

of waterf; and therefore the numerical value of h could be

given for water only. Regnault has since extended his

measurements to other fluidsJ,and it is now possible to apply

the equation to obtain the numerical value of h for these

fluids. We thus obtain the following results.

Bisulphide of Carbon : CSjj. According to Regnault we

have

[ cdt = 0-23523« + O'OOOOSlSi',

Jo

* Eankine's paper was read tefore the Eoyal Society of Edinburgli in

February, 1850, and printed in the Transactions, Vol. xx. p. 147. The author's

paper was read before the academy of Berlin ill February, 1850, and printed

in Poggendorf's Annalen, Vol. lxxix. pp. 368 and 500.

t Relalions des Eaeiperienceii, Vol. i., Paris, 1847.

J Belations des Experiences, Vol. ii., Paris, 1862.




r +Jo

90 + OUeOlt - 00004123^

whence we have

c = 0-23523 + 00001630«,

r = 90-00 - 0-08922«- 00004938^.

Substituting these Values, equation (25) becomes

, «««ô^.^. 9000-0-08922«-00004938<''h = 0-14601 - 0-0008246<

273+T'

hence we obtain for h the following values amongst others

t




In the case of Ether therefore the specific heat of

saturated steam has, at least at ordinary temperatures, a posi-

tive value.

Chloroform : CHCI3. According to Regnault we have

fcd< = 0-23235« + 0-0000507a«',

Jo'

r+l cdt = 67 + 0-1375«

whence we have

c =- 0-23235 + 0-00010144<,

r = 67 - 0-09485f - 000005072«'.

Equation (25) thus becomes

I ^ n o.r« 67 - 0-09485* - 0-00005072i(''

h^ 01375273T"^

'

and from this the following values are deduced

t




and from this the following values are deduced

t




In all the formulae for h given above we see that its value

increases as the temperature rises. In the only case, that of

ether, in which it is positive at ordinary temperatures, its

absolute value increases as the temperature rises. In theother cases, in which it is negative, its absolute value

diminishes; it thus approaches to zero, and it would appear that

at some higher temperature it would attain the value zero,

and at stiU higher temperatures would become positive. Todetermine the temperature at which A = 0, we have by equa-

tion (25)

^+c-J=0 (32).

In this equation we must, as above, express c and r as

functions of t, and then solve it with regard to t.

The empirical formulis of Regnault, by means of which

we have expressed c and r as functions of t, should not of

course be applied much beyond the limits of temperature

within which Regnault carried out his experiments. Hence

the determination of the temperature for which A. =is in many cases impossible, as for instance with water,

where the equations obtained by putting A = in (29)

and (31) would lead to a value for t of about 600", where-

as the equations are only applicable up to somewhat over

200". But with other fluids the temperature for which

h = Q, and above which h is positive, lies within the limits of

application of the formulae. Thus Cazin* calculates thistemperature for Chloroform at 123"48°, and for Bi-chloride of

Carbon at 128-9°.

§ 5. Specific Heat of Saturated Steam, as proved by

Expeninent.

The result arrived at by theory, that the Specific Heat of

saturated steam is negative, and that therefore saturated

steam, if expanded in a non-conducting envelope, must

partially condense, has since been experimentally proved by

Hirn-)-. A cylindrical vessel of metal was fitted at the two

ends with parallel plates of glass, so that it could be seen

through. This, when filled with steam at high pressure was

* Annates de Chimie et de Physique, Series iv. Vol. xiv.

t Bulletin 133 de la Soci4U Industrielle de Mulhmise, p. 137.




perfectly cldar; but when a cock was suddenly opened, so

that part of the steam escaped, and the remainder expanded,

a thick cloud appeared in the interior of the cylinder, proving

a partial condensation of the steam. Subsequently, whenVolume II. of Renault's Relation des Eceperiences had ap-

peared, containing the data, given above, to determine h for

other fluids, and shewing that for ether h must be positive. Himproceeded to experiment with that vapour also. His description

is as follows * :" To the neck of a strong crystal flask I

connected a pump, the capacity of which was nearly equal to

that of the flask, and which was provided with a cock at the

bottom. Some ether was poured into the flask, and it wasimmersed to the neck in water at about 50°. The cock was

then kept open, until all the air was assumed to be expelled.

Then the cock was closed, and the pump plunged into the hot

water with the flask: whereupon the ether vapour raised

the piston to the top. The apparatus was now suddenly

taken from the water, and the piston forced rapidly down.

At this moment, but for a moment only, the flask became

filled with a distinct cloud." It was thus shewn that ether

vapour behaves conversely to steam, partially condensing, not

during expansion, but during compression; a fact which

is in accordance with the opposite sign of h in the two

cases.

To check this conclusion Him made an exactly similar

experiment with Bi-sulphide of Carbon. The result was

that on forcing down the piston the flask remained per-fectly transparent. This is again in accordance with the

theory, since with Bi-sulphide of Carbon, as with water,

h is negative, and compression of the vapour produces a rise

of temperature, and not a fall. Some years later Cazin"f,

aided by the Association scientifique, made with great care

and skill a similar series of experiments, in some respects

more extended. He used as before a cylinder of metal, fitted

with glass at the ends. This was placed in a bath of oil, so

as to give it the exact temperature proper for the experi-

ment. The first series of experiments embraced only the

expansion of steam; the arrangement was such that, whenthe cylinder was filled with vapour, a cock could be opened,

* Cosmos, 10 April, 1863.

+ Anwilcs de Chimie et de J^hjfsique, Series it. Vol, xit.




through which a part of the vapour escaped either into the

atmosphere or into an air vessel, the pressure in which

could be kept at any given point below the pressure of the

vapour. In a second series of experiments a

pump wasconnected with the cylinder; this was placed in the same

bath of oil, and the piston could be moved ra^dly back-

wards or forwards by specigj. mechanism, so as to increase or

diminish the volume of the vapour.

By these experiments the results obtained by Hirn for

steam and ether were confirmed, and with the second ap-

paratus a double proof was given in each case, viz. both by

rarefaction and condensation. Steam formed a cloud duringrarefaction, whilst it remained quite clear during condensa-

tion. Ether, on the contrary, formed a cloud during con-

densation, and remained clear during rarefaction. Somespecial experiments were further made with vapour of chloro-

form. As mentioned above, in the case of chloroform A,

which is negative at lower temperatures, becomes zero at a

temperature which Cazin has calculated at 123'48°, and at

still higher temperatures is positive. This vapour must thus

partially condense during expansion at lower temperatures,

and must partially condense during compression at higher

temperatures beyond the point of transition. With the first

apparatus, which only allowed of expansion, clouds were

observed during expansion at temperatures up to 123°. At

temperatures above 145° there was no formation of cloud.

Between 123° and 145° the conditions were somewhat different

according to the degree of expansion. With a small degree

of expansion there was no cloud ; with a higher degree some

formation of a cloud appeared towards the end of the pro-

cess. The explanation of this is simple. The high expansion

had produced a considerable fall of temperature, and the

vapour had thereby been reduced to the temperature at

which expansion is accompanied by a fall of tempera^

ture. The result is thus completely in accordance with the

theory. With the second apparatus the vapour of chloroform

formed a cloud during expansion up to 130°, whilst it re-

mained perfectly transparent during compression. Above

136° a cloud was formed during compression, whilst it re-

mained clear during expansion. The theory is hereby more

fully established than by the first apparatus. The circumstance



142 ON THE MECHANICAL THEORT OF HEAT.

that the temperature at which the law of the vajpour changes

appeared in these experiments to lie between 130° and 136°,

whilst theory gives it at 123-48'', is not a matter of great im-

portance. On the one hand these experiments are not adapted

for an accurate determination of this temperature, because

they always involve finite changes of volume of considerable

magnitude, whereas the theory embraces indefinitely small

changes only. On the other hand, Cazin hiriiself mentions

that his chloroform was not chemically pure, and required for

a given vapour-pressure a higher temperature than that

found by Eegnault. Having regard to these circumstances,

the theory must be considered as being fully confirmed byexperiment.

§ 6. Specific Volume of Saturated Vapour.

We will now consider the second of the two quantities

mentioned at the beginning of § 2, viz. s, or the specific volume

of the saturated vapour.

It was formerly the custom to use the law of Mariotte and

Gay-Lussac, in order to calculate the volume which a gas as-

sumes under difierent conditions of temperature and pressure,

and to take no account of whether the vapour was in the satu-

rated or superheated condition. It is true that from manyquarters doubts were expressed, whether vapours really fol-

lowed this law up to the saturation point: but, as the experi-

mental determination of the volumes offered great difficulties,

and a theoretical determination was impossible from the want

of well-established principles, it remained the custom to

apply the above law in this case, so as at least to arrive

at some sort of determination of the volume of saturated

vapour. But the' equations obtained by the author, and

given at the end of § 1, now offer us a means of arriving at

a strict theoretical calculation for the volume of saturated

vapour, which, when the data are given, may be worked outin practice. For in these equations occurs the quantity u,

which =s — a; where a is the specific volume of the fluid.

This, as a rule, is very small in comparison with s, and

may be neglected in many calculations ; but it is still a

known quantity, and may be taken account of without diffi-

culty.



SATURATED VAPOUK. 143

Substituting (s—a) for u in the last of these equations, (17),

we obtain

T(s-a) dp .

^ ^~~ ^dT

^•^^^'

or, solving the equation for s,

Ers=-=f- + <r {U).

IT

By this equation the specific volume of the saturated va-pour may be calculated for all substances, whose pressure pand heat of vaporization r are known as functions of the

temperature,

§ 7. Departurefrom the law ofMariotte and Gay-Lussac

in the case of Saturated Steam,

We will first apply the foregoing equations to ascertain

whether saturated steam follows the law of Mariotte and Gay-

Lussac, or how far or in what way it departs from it.

If it follows the law the following equation must hold

^= const.,

or, substituting a + f for T, and multiplying by ^,

1 a,

TV Ps 7 = const.U^ a + t

but from equation (33), substituting a + < for T, we obtain

y,p{s-a)---= =-T- (35).

^ 'pat

As the difference [s -> a) differs little from », the left-band

side of these two equations is very nearly the same, and, to





SATURATED VAPOUK. 145

t in Centi-




the modulus of that system. By the help of these values of

- -'^- and of the values of r given by the equation stated

p dt

above, and lastly of the value 273for a, the values were

calculated which the expression on the right-hand side of

equation (35) and therefore likewise the expression

1 - a

assumes for the temperatures 5°, 15", 25°, etc. These values are

given in the second column of the table below. For tempe-

ratures over 100° the two series of numbers found above for pwere both made use of, and the two results thus obtained are

placed side by side. The meaning of the third and fourth

columns will be more fully explained below.

1.



SATUEATED VAPOUK. 147

1

This table at once shews that ^ « (s — <r)—- is not constant,

as it must be if the law of Mariotte and Gay-Lussac holds,

but decreases decidedly as the temperature rises. Between35° and 95° this decrease appears very regular. Below 35°

the decrease is less regular, the simple explanation being that

here the pressure p and its differential coefficient -^ are very

small, and therefore small errors in their determination, which

are quite within the limits of errors of observation, may yet

become relatively important. Above 100° the values of the

expression are not so regular as between 25° and 95°, but

shew on the whole a similar rate of decrease : and if wemake a graphic representation of these values, it is found

that the curve, which below 100° passes exactly through the

points determined by the numbers contained in the table,

can be readily continued up to 230° in such a way that these

points are distributed equally on both sides of it.

The course of this curve can be expressed with sufficient

accuracy for the whole extent of the table by an equation of

the form

^P(«-'^)î = "*-»'«** (37),

where e is the base of Napierian logarithms, and m, n, k are

constants. If we determine the latter from the values which

the curve gives for 45°, 125°, and 205°, we obtain

m = 31-549; w = 1-0486; ^ = 0-007138 (37a),

and if for convenience we use common logarithms, we finally

obtain

Logr31-549--^.p(5-o-)a-ht

= 0-0206 + 0003100«... (38),

The numbers contained in the third column of the table

are calculated from this equation, and in the fourth column

are given the differences between these and the nUmbers in

the second cplumn,

10—2



148 ON THE MECHAlflCAL THEORY OF HEAT.

§ 8. Differential Coefficienta of—

Tiie foregoing analysis leads easily to a formula, from

whick we can ascertain more exactly the mode in which

steam departs from the law of Mariotte and Gay-Lussac,

Assuming, this law to hold, we shall be able to put

PI: a + t

where ps^ representsthe value of ps at 0". The differential

coefficienit -,-( — ) would then have a constant value, viz. the

at \psgj

well-known coefficient of expansion - = 0003665, Instead

of this, equation (37) gives, if we use s -in place of (s — tr),

the equation

Fô

me''*

a + tX , (39).

whence

d /ps_

dt [ps^

1 m - n [I + k (a + t)] e''t ,.„= - X — (40).a m — n

Thus the differential coefficient is not a constant, but a

function which decreases as the temperature increases. This

function, if we substitute for m, n, and Ic the numbers given

in (37a), has amongst others the following values, for different

temperatures.

t.

0«



SATURATED VAPOUR, 140

From the above table we see that it is only at low tem-

peratures that the variations from the law of Mariotte and

Gay-Lussac are small, and that at higher temperatures, e.g.

above 100°, they can by no means be neglected.A glance at the tabl-e is sufficient to shew that the values

found for -n (—

)are smaller than 0-003665

-, whereas it is

at \psj

known that for the gases which vary considerably from the law

of Mariotte and Gay-Lussac,such as carbonicacid and sulphuric

acid, the coefficient of expansion is not smaller, but greater

than the above number. Hence this differential coefficient

cannot be taken to correspond with the coefficient of expansion

which relates to increase of volume by heating at constant

pressure, nor yet with the figure obtained if we leave the

volume constant during the heating, and then observe the

increase of the expansive force. Thus we have here a third

special case of the general differential coefficient -jii ) > viz.

that in which the pressure increases during the heatingunder the same conditions as in the case of steam, when this

retains its maximum density; and this case must be con-

sidered for carbonic acid likewise, if we are to establish a com-

parison.

Steam has at about 108°an expansive force equal to 1 metre

of mercury, and at 129J° equal to 2 metres. We will examine

what takes place with carbonic acid, if this is also heated by

21^°, and thereby the pressure increased from 1 to 2 metres.

Regnault* gives as the coefficient of expansion of carbonic

acid at constant pressure 0'00371 if the pressure is 760 mm.,

and 0"003846 if the pressure is 2520 mm. For a pressure of

1500 mm., the mean between 1 and 2 metres, the coefficient

of expansion, if assumed to increase in proportion to the

pressure, will have the value '003767. If carbonic acid

under this mean pressure is heated from 0° to Sl^'", the

quantitv — wiU increase from 1 to

1 + 0-003767 X 21-5 = 1-08099.

Again other experiments of Regnault'sf have shewn that, i|

* Relation d's Experiences, t. i. Mem. 1.

t Relation des Eny^rienees, 1. 1. Mem. 6.




carbonic acid, which had a pressure of about 1 m. at a tem-

perature of aiiout 0°, is loaded with a pressure of 1 '98292 m.,

the quantity pv decreases in the ratio of 1 : 0'99146 ; or for an

increase of pressure from1

to 2ms., it will decrease in the

ratio of 1 : 0'99131. Now if both of these take place

together, viz. a rise of temperature from 0° to 21^°, and

a rise of pressure from 1 to 2 ms., then the quantity ^

must increase from 1 to 108099 x 0-99131 = 1-071596 very

nearly, whence we obtain as the mean value of the differen-

d / pv^

tial coefficient „,

at \pi\.

^•Q^f^ ^ 000333.21-5

We thus see that for the case here considered we have a

value for carbonic acid which is less than 0003665, and

there is therefore less, ground for surprise at obtaining the

same result for steam at its maximum density:

If we seek to determine on the other hand the actual

coefficient of expansion for steam^ or the number which

shews how far a quantity of steam expands, if it is taken at

a given temperature and at its maximum density and then

heated, apart from water, at constant pressure, we shall cer-

tainly obtain a value which is larger, and probably much

larger, than 0-003665.

§ 9. Formula to determine the Specifia Volume ofSatu-

rated Stea/m, and its comparison with experiment.

From equation (37), and equally from equation (34), the

relative values of s — a; and therefore to a close approxima-

tion those of s, may be calculated for different temperatures,

without needing to know the Mechanical Equivalent E. If

however we wish to calculate from the equations the absolute

value of s, we must either know E^ or must attempt to

eliminate it by the help of some other known quantity. Atthe time when the author first began these researches, several

values of E had been given by Joule, taken from various

methods of experiment: these differed widely from each

other, and Joule had not announced which he considered the




most probable. In this uncertainty the author determined

to attempt the determination of the absolute value of s from

another starting point, and he believes that his method still

possesses interest enough to merit description.

The specific weight of gases and vapours is generally ex-

pressed by comparing the weight of a unit of volume of the

gas or vapour with the weight of a unit of volume of at-

mospheric air at the same pressure and .temperature. Simi-

larly the specific volume may be expressed by comparing

the volume of a unit of weight of the gas or vapour with the

volume of a unit of weight of atmospheric air at the same

pressure and temperature. If we apply this latter method tosaturated steam, for which we have denoted the volume of

a unit of weight by s, and if we designate by v' the volume

of a unit of weight of atmospheric air at the same pressure

and temperature, then the quantity under consideration is

given by the fraction — .

For s we have the following equation, obtained from (37)

by neglecting <r :

gÊ(a+J)^

For v' we have by the law of Mariotte and Gay-Lussac

the equation

, jj, a +V =Jti .

P

These two equations give

-, = „-(™ — «e*') (42).

If we form the same equations for any given temperature

and

obtain

t^, and denote the corresponding value of - by (- j , we

[->) = p7-(TO-«e*'°).




If by the help of this equation we eliminate the constantp

factor T=rr- from (42), we obtainKa

t^(s_\ m-n^ ^^g.

The question is now whether, for any given temperature

t„, the quantity f-j or its reciprocal (- ) , which expresses

the specific weight of the steam at temperatute t„, can be

determined with sufficient certainty.

The ordinary values given for the specific weight of steam

refer not to saturated but to highly superheated steam.

They agree very well, as is known, with the theoretical

values which may be deduced from the well-known law as

to the relation between the volume of a compound gas, and

those of the gases which compose it. Thus e.g. Gay-Lussac

found for the specific weight of steam the experimental value

0"6235; whilst the theoretical value obtained by assuming

two units of hydrogen and one unit of oxygen to form, by

combining, 2 units of steam, is

2 X 006926 X 1-10563 „^„„

^= 0-622.

This value of the specific weight we cannot in general

apply to saturated steam, since the table in the last section,

which gives the values of -ri ] < indicates too large a

divergence from the law of Mariotte and Gay-Lussac. Onthe other hand the table shews that the divergences are

smaller as the temperature is lower; hence, the error will

be insignificant if we assume that at freezing temperature

saturated steam follows exactly the law of Mariotte and Gay-

Lussac, and accordingly take 0-622 as the specific heat atthat temperature. In strict accuracy we must go yet further

and put the temperature, at which the specific weight of

saturated steam has its theoretical value, still lower than

freezing point. But, as it would be somewhat questionable

to use equation (37), which is only empirical, at such low

temperatures, we shaU content ourselves with the above



SATURATED VAPOUH. 153

assumption. Thus giving to t^ the value 0, and at the same

time putting (-) = 0-622 and therefore (-)\ = jr^ , we ob-\« /o \v Jo 0622

tain from equation (43)

m — weM.(44).

v 0-622 (to- w)

From this equation, using the values for m, n, and h

given in (37a), the quantity -7, and therefore the quantity s,

may be calculated for each temperature. The foregoingequa,tion may be thrown into a more convenient form byputting.

V.(45),

and by giving to the constants M, iV, and a the following

values, calculated from thoseof

to, n,

and k:

-3f= 1-663; iV=0-05527,- a = 1-007164.. .(45a).

To give some idea of the working of this formula, we give

in the following table certain values of -, and of its re-V

. v *

ciprocal -, which for the sake of brevity we'stall denote by

the letter d, already used to designate specific weight.

t




judges. The author believes however that it is now generally

accepted as correct.

It has also received an experimental verification by the

experiments of Fairbairn and Tate*,published in 1860. The

results of their experiments are compared in the follow-

ing table, on the^ one hand with the results previously ob-

tained by assuming the specific weight to be 0-622 at all

temperatures, and on the other hand with the values cal-

culated by equation (4-5)'.

Temperature

in Degrees

Centigrade.




perimental values are in general yet further removed from

those previously obtained than are the values derived from

the author's formula.

§ 10. Determination of the Mechanical Equivalent ofHeatfrom the behaviour of Saturated Steam.

Since we have determined the absolute values of s, •with-

out assuming the mechanical equivalent of heat to be known,

we may now applj' these values, by means of equation (17),

to determine the mechanical equivalent itself. For this

purpose we may give that equation the following form

£=—T^Cs-.') («).

The coefficient of s — o- in this equation may be calculated

for different temperatures by means of Regnault's tables. For

example, to calculate its values for 100", we have given for

-ij the value 27"20, the pressure being reckoned in milli-

metres of mercury. To reduce this to the measure here em-

ployed, viz. kilogrammes per square metre, we must multiply

by the weight of a column of mercury at temperature 0",

1 square metre in area and 1 millimetre in height, that is by

the weight of 1 cubic decimetre ofmercury at 0". As Regnault

gives this weight at 13"o96 kilogrammes, the multiplication

gives us the number 369 "8. The values of {a + t) and of r

at 100° are 373 and 536'5 respectively. Hence we have

/ a. rt ^£^"' "^

' ~dt _ 373 X 369-8 _ _.^

;: 536-5"^''^'

and equation (46) becomes

^=257(s-o-) (47).

We have now to determine the quantity (s — a), or, since o- is

known, the quantity s for steam at 100°. The method formerly

pursued, i.e. to use for saturated steam the same specific

weight, which for superheated steam had been found by

experiment or deduced theoretically from the condensation of




water, led to the result, that a kilogram of steam at 100°

should have a volume of 1'696 cuhic metres. From the fore-

going however it appears that this value must be consider-

ably too large, and must therefore give too large a value for

the mechanical equivalent of teat. Taking the specific heat

as calculated by equation (45), which for 100° is 0"645, we

obtain for s the value 1'638. Applying this value of s we

get from equation (47)

^ = 421 ....(48).

This method therefore gives for the mechanical equivalent

of heat a value which agrees in a very satisfactory manner

with the value found by Joule from the friction of water, and

with that deduced in Chapter II. from the behaviour of

gases ; both of which are about equal to 424. This agree-

ment may serve as a verification of the author's theory as to

the density of saturated steam.

§ 11. (complete Differential Equationfor Q m the case of

a body composed both of liquid and vapour.In § 1 of this chapter we expressed the two first differ-

ential coefficients of Q, for a body consisting both of liquid

and vapour, by equations (7) and (8), as follows:

dm ^'

^ = m{E-G) + MaHence we may forai the complete differential equation of the

first order for Q, as follows

dQ = pdm + o].{n-G) + MG\dT., (49).

By equation (12) we may put

TT p_dp P

whence equation (49) becomes

dQ=pdm+ m(^^-^ + Ma dT (50).



satueated vapour. 157

Since p is a function of T only, and therefore

we have dQ = d{tnp) + {-'^ + MC\dT (51),

or dQ = Td{^^^+MCdT (52).

These equations are not integrable if the two quantities, whose

dififerentials are on the right-haad side, are independent of

each other, and the mode of the variations thus left undeter-

mined. They become integrable as soon as this mode is deter-

mined in any way. We can then perform with them calcu-

lations exactly similar to those given for gases in Chapter II.

We will for the sake of example take a case which' on

the one hand has an importance of its own, and on the other

derives an interest from the fact that it plays a prominent

part in the theory of the steam-engine. The assumption is

that the mass consisting both of liquid and vapour changes

its volume, without any heat being imparted to it or taken

from it. In this case the temperature and magnitude of the

gaseous portion also suffers a change, and some external

work, positive or negative, must at the same time be per-

formed. The magnitude m of the gaseous portion, its volume

V, and the external work W, must now be determined as

functions of the temperature.

§ 12. Change of the Gaseous Portion of the Mass.

As the mass within the vessel can neither receive nor

give off any heat, we may put dQ = 0. Equation (52) then

becomes:

Td (^-P^+MCdT=0 (53).

If we divide this equation by E, the quantities p and C, which

relate to the mechanical measure of heat, change into r and

c, which relate to the ordinary measure of heat. If we also

divide the equation by T, it becomes:

dl-jrj+Mc-^=0 (53a). .,




The first member of this equation is a simple differential,

and may at once be integrated : the integration of the second

is also always possible, since c varies only with the tempera-

ture T. If we merely indicate thisintegration,

and denotethe initial values of the various magnitudes by annexing

the figure 1, we obtain the following equation:

"^^^-mQ^ (54).

mr m,r, ,, f^ dTor

Actually to perform the integration thus indicated, we

may employ the empirical formula for c given by Regnault,

For water this formula, already given in (27), is as follows

c = 1 + 000004 + 00000009*'.

Since c is thus seen to vary v6ry slightly with the tempera-ture, we will in our calculations for water assume c to be

constant, which will not seriously affect the accuracy of the

results. Hence (54) becomes:

~^=-^-Mclog-^ (oo),

whence

m =f(^-^clogJj (56).

If we here substitute for r the expression given in (28), or in

a simpler form in (30), then m will be determined as a func-

tion of tenaperature.

To give a general idea of the values of this function,

some values have been calculated for a special case, and col-

lected in the following table. The assumption is that thevessel contains at first no water in a liquid condition, but

is filled with steam at its maximum density, so that in

equation (56) we may put m^ = M. Let there now be anexpansion of the vessel. A compression would not be ad-

missible, because on the assumption of the absence of water

at the commencement, the steam would not remain at its





160 ON THE MECHANICAL THEORY OF HEAT."

The following table gives a series of values of the quotient

— , calculated by this equation for the same case as was treated

in the last table. Under these are placed for the sake of

comparison the values of — , which would hold if the two

ordinary assumptions in the theory of the steam-engine were

correct : viz. (1) that steam in expansion remains at its maxi-

mum density without any part of it condensing; (2) that

steam follows the law of Mariotte and Gay-Lussac. On these

assumptions we shall have

.t



SATUEATED VAPOUK. 161

In this equation we may substitute for mu -— the expres-

sion given in equation (57), and may then perform the inte-

gration. The result however is obtained in a more convenient

form as follows. By (13) we have:

and from (53) we obtain

hence

'^dT= d(mp) + MCdT;

mu^dT=d (mp) + MCdT.

Equation (60) now becomes

pdv=d{mup) — d (mp) —

MCdT= -d[m(p-up)]-MGdT (61).

Integrating this equation we obtain

W= m;(p^-u^;)-m(p-wp) + MC{T^-T)...(Q2).

If in this equation we substitute for p and (7, according to

equation (14), the values JEr and JEc, and collect together theterms which contain ^ as a factor, we have

W= mup - m^^p^ +E [m^r^-mr+ Mc{T^- T)]... (63).

From this equation we may calculate W, since mr and

mu are already known from the equations (56) and (58). This

calculation has been made for the same case as before, and

Wthe values of -jr, i.e. of the work done by a unit of weight

during expansion, are given in the following table ; the unit

of weight is here a kilogram and the unit of work a kilo-

grammetre. The value used for H is 42355, as found by

Joule.

As a comparison with the numbers of this table it may be

c. 11




again mentioned that the value obtained for the work done

during the actual formation of steam, as this overcomes the

external pressure, is 18700 kilogrammetres per kilogram of

water, evaporated at temperature 150° and at the correspond-

ing pressure.

t



CHAPTER VII.

FUSION AND VAPORIZATION OF SOLID BODIES.

§ 1. Fundamental Ilqiiationsfor the process of Fusion.

Whilst in the case of vaporization the influence of the ex-

ternal pressure was eaxly observed, and was everywhere taken

into account, it had hitherto been left out of account in the

case of fusion, where it is much less easily noticed. A little

consideration however shews that, if the volume of a body

changes during fusion, the external pressure must have

an influence on the process. For, if the volume increases, an

increase of pressure will make the fusion more difficult,

whence it may be concluded that a higher temperature is

necessary for fusion at a high than at a low pressure. If on

the other hand the volume decreases, an increase of pressure

will facilitate the fusion, and the temperature required will

be less, as the pressure is greater.

To examine more exactly the connection between pressureand fusion-point, and the peculiar changes which are some-

times connected with a change of pressure, we must form the

equations which are supplied for the process of fusion by the

two fundamental principles of the Mechanical Theory of

Heat. For this purpose we pursue the same course as for

vaporization. We conceive an expansible vessel containing

a certain quantity iHf of a substance, which is partly in the

solid, and partly in the liquid condition. Let the liquid part

have the magnitude m, and therefore the solid part the mag-

nitude M—m. The two together are suôsed to fill the

vessel completely, so that the content of tffi; vessel is equal

to V, the volume of the body.

If this volume v and the temperature T are given, thie

magnitude m is thereby determined. To prove this, let us

11—2




first suppose that the body expands during fusion. Let it be

also in such a condition that the temp'ei-ature T is exactly the

melting temperature at that particular pressure. Now if in

this condition the magnitude of the liquid part were to

increase at the expense of the solid, the expansion which

must then result would produce an increase of pressure

against the walls of the vessel, and therefore an increased

reaction of the walls against the body. This increased pres-

sure would produce a rise in the fusion-point, and since the

existing temperature would then be too low for fusion, a

solidification of the liquid part must begin. If on the con-

trarythe

solid partwere

to increase atthe

expense ofthe

liquid, the point of fusion would thereupon sink, and since

the existing temperature would then be higher than the

fusion-point, a fusion of the solid part must begin. Next, if

we make the opposite assumption, viz. that the volume

decreases during fusion, then if the solid part increase there

must be a rise of pressure and in consequence a partial

melting, and if the liquid part increase there must be a fall

in pressure and in consequence a pai-tial solidification. Thuson either assumption we have the same result, viz. that only

the original proportions of the liquid and solid parts (which

proportions correspond to the pressure which gives a tem-

perature of fusion equal to the given temperature) can be

permanently maintained. Since then the magnitude m is

determined by the temperature and volume, this volume will

conversely be determined by the temperature and the mag-

nitude m; and we may choose T and m as the variables

which serve to determine the condition of the body. It now

remains to express ^ as a function of T. Here we mayagain apply -equations (1), (2), (3) of the last chapter, viz.

dT [dm] dm \dTJ '^'dT^d^'

.d_ fdQ\ d_ fdQ\ _ jL^ dQdT\dm) dm\dT) T^dm'

iQ^rpdp ^ dv_

dm dT dm

If we denote by o-, as before, the specific volume (or volumeof a unit of weight) for the liquid condition of the body,



FUSION AND VAPORIZATION OP SOLID BODIES. 165

and the specific volume for the solid condition by t, we havefor the total volume v of the body,

V = ma-

+{M —

m)t,

or v=m{a--T) + MT (1),

^^ê^°e ^='^-'^ (2).

If further we denote the heat effusion for a unit of weight by

p', we may put

d^-p (^^-

To express -^, the other differential coefficient of Q, we

must employ symbols for the specific heat of the body in the

liquid and in the solid condition. Here, however, we must

make the same remark as in the case of vaporization, viz.

that it is not the specific heat at constant pressure which is

treated of, but the specific heat for the particular case in

which the pressure alters with the temperature in such a way

that the temperature shall always be the temperature of

fusion for that particular pressure. In the case of vaporiza-

tion, where the changes of pressure are generally small, it

was possible to neglect the influence of the change of pressure

on the specific heat of the liquid body, and to consider the

specific heat of a liquid body, as found in the formula, to be

equivalent to the specific heat at constant pressure. In the

present case small changes of temperature produce such

great changes of pressure, that the influence of these oq the

specific heat must not be neglected. We will, therefore, under

the present circumstances, denote by 0" the specific heat of

the liquid, which in the formula for vaporization we denoted

byC. The specific heat of the solid body may be de-

noted in this case by K'. Applying these symbols we may

write

'^ = mC' + {M-m)K',

^=m{0'-K')+MK' (4).




From equations (3) and (4) we have

dT\dm)~dT^

'•

U^)-^-""'(^)-

,dm \<

Inserting these sralues, and the ralue for -^ given in (3), in

the above differential equations, we obtain

%+,K'-0' = i.-r)^ (7),

%-^^'-^'4^'\

/ = T(<r-r)^ (9).

In these equations the heat is supposed to be measured by

mechanical units. If the heat is to be measured in ordinary

units, we may use the following sj'mbols :

The equations then become

£+'--«'-^'© <">

g,+ A'-c' =J (12),

T(a--T) fdp\'=-^® ('«)

These are the equations required, of which the first corre-

sponds to the first Fundamental Principle, and the second to

the second, whilst the third is a combination of the other

two.



FUSION AND VAPOEIZATIOK OF SOLID BODIES. 167

§ 2. Relation between Pressure and Temperature ofFicsion.

The foregoing equations, only two of -which are inde-

pendent, maybe applied to determine two quantities hitherto

unknown.

We will first use the last equation to determine the wayin which the temperature of fusion depends on the pressure.

The equation may be written

^T Tj.-r)

dp Er'^^*>-

This equation in the first place justifies the remark already

made, that if a body expand during fusion the point of fusion

rises as the pressure increases ; whereas if the body contracts

the point of fusion falls. For according as cr is greater or

less than t so is the difference a-— t, and therefore also the

dTdifferential coefficient -^ .. positive or negative. Again, by

this equation we may calculate the numerical value of —i- .

We will calculate this value for the case of water. The vo-

lume in cubic metres, or the value of <r, for a kilogramme of

water at 4° C. is O'OOl. At freezing point it is a little greater,

but the difference is so small that it may be neglected. The

volume in cubic metres, or the value of t, for a_ kilogram

of ice is 0001087. The heat of fusion for water, or thevalue of r, is according to Person 79. At freezing point Tequals 773, and for E we wiU take the value 424. Hence

we obtain

dT_ 273 X 0-000087

dp"

424 X 79

If the pressure is given not in mechanical units (kilograms

per square metre), but in atmospheres, we must multiply the

above value of -j- by 10333. This gives us

^=-000733,d/p

i.e. an increase of pressure of one atmosphere will lower the

point of fusion by G'00733 of a degree Cent.



168 ON THK MECHANICAL THEORY OF HEAT.

§ 3. Experimental Verification of the Foregoing Result.

The conclusioii that the melting point of ice is lowered

by an increase of pressure, and the first calculation of the

amount, are due to James Thomson, -who derived from

Gamot's theory an equation -which differs from our equation

T(14) only in tliis, that in the place of -p it contams an un-

known function of temperature, whose particular value for

the freezing point was determined from Regnault's data on

the heat of vaporization andpressure of steam. Sir William

Thomson afterwards applied to thistheoretical result a very

accurate test by experiment*.

In order to measure small differences of temperature, he

prepared a thermometer filled with ether-sulphide, the bulb

of which was 3^ in. long and the tube 6J in. Of this 5^ in.

were divided in 220 equal parts, and 212 of these parts com-

prised an interval of temperature of 3° Fahr., so that each

part was about equal to ^ of a degree Fahr. This thermo-

meter was hermetically enclosed in a larger glass tube, to

protect it from the external pressure, and so enclosed was

placed in an Oersted press, filled with water and lumps of

clear ice, and containing an ordinary air gauge to measure

the pressure. When the thermometer had become stationary

at a point corresponding to the melting point of ice at atmos-

pheric pressure, the pressure was increased by screwing do-wn

the press. The thermometer was at once seen to fall, as the

mass of water and ice assumed the lower melting tempera-

ture corresponding to the higher pressure. On taking off,

this pressure the thermometer returned to its original position.

The table below gives the fall of temperature observed for two

Pressure.



FUSION AND VAPORIZATION OF SOLID BODIES. 169

dTas high as 16"8 atm. the value of t— , which was found in the

dp

last section, and which primarily relates to the ordinary at-

mospheric pressure.

We see that the observed and calculated numbers agree

very closely together, and thus another result of theory

has been verified by experiment.

More recently a very striking experiment was performed

by Mousson*, who by the application of enormous pressures

melted ice which was kept during the experiment at a

temperature of — 18° to — 20°. The pressure employed he

calculates approximately at about 13000 atmospheres ; onwhich it may be remarked that it may be possible to pro-

duce the melting with a much smaller pressure, since with

his arrangement all that could be known was that the ice

had somehow melted during the experiment, and not the

exact time at which the melting took place.

§ 4. Experiments on Svbstances which eoopand during

Fusion.

Bunsent was the first to institute experiments on sub-

stances which expand during fusion, and of which the

fusion point must therefore rise as the temperature in-

creases. The substances he chose were spermaceti and

paraffin. By an ingenious arrangement he obtained in an

extremely simple manner a very high and at the same time

measurable increase of pressure, and was able to observe

portions of the same substance side by side under ordinary

atmospheric, and under the increased pressure. He took a

tube of thick glass about the size of a straw and 1 foot in

length, and drew it at one end into a capillary tube 15 to

20 inches long, and at the other end into a somewhat larger

one only 1J inches long. The latter, which was placed lowest

in the apparatus, was bent round until it stood up parallelto the lower part of the glass tube. This short curved part

was filled with the substance to be tested, and the larger

glass tube with quicksilver, whilst the long capillary tube

remained filled with air. Both capillary tubes were sealed

* Pogg. Ann., Vol. cv. p. 161.

t Pogg. Ann., Vol. lxjcxi. p. 5G2.



170 ON THE MEtiHANICAL THEORY OF HEAT.

at the ends. On heating the apparatus the quicksilver

expanded, rose in the longer capillary tube, and compressed

the air within it. The reaction of this air compressed first

the quicksilver and then the substance in the shorter tube,

and the magnitude of the pressure, which was capable of

rising to above 100 atmospheres, could be measured by the

volume of air left in the upper tube.

This apparatus was fixed on a board close to another

arranged in the same way except that the upper air-tube

was not sealed ; so that no compression of the air, and con-

sequent rise of pressure, could take place. The two tubes

were now plunged in water, the temperature of which wassomewhat higher than the melting point of the substance

to be tested. Thus when the lower tube filled with the

substance was once completely under water, it was only

needed to sink it still deeper in order to heat a larger part

of the quicksilver, and so to obtain a higher pressure

in the closed upper tube. Under these conditions Bunsen

repeatedly melted the substance in both tubes, and then by

cooling the water allowed it again to solidify, observing the

temperature at which this took place. The result was that

this solidification always took place at a higher temperature

in the tube in which the pressure was increased than in^

the other. The following were the numerical results.

Spermaceti.



FUSION ANB VAPOEIZATION OF SOLID BODIES. 171

More recently Hopkins* experimented with spermaceti,

wax, sulphur, and stearine, producing pressures by means of

a weighted lever to 800 atmospheres and upwards. With all

the above substances a rise of the melting point under an

increase of pressure was observed. The particular tempera-

tures observed with different pressures shewed however

considerable irregularities. In the case of wax, with which

the rise of temperature was most regular, an increase of

pressure of 808 atmospheres produced a rise in the melting

point of 15J° Cent.

The calculation of the rise in the melting point from

the theoretical formula cannot well be performed for thesubstances tested by Bunsen and Hopkins, since the data

required are not known with sufficient accuracy.

§ 5. BddUon between the h$at oonswmed in Fusion and

the temperature of Fusion.

Having applied equation (13), in § 2, to determine the

relation between the temperature of fusion and the pressure,

we will now turn to equation (12), which may be written as

follows

g=C'-A'+^y (15).

This equation shews that, if the temperature of fusion is

changed by a change of pressure, the quantity of heat r

required for fusion also changes. The amount of this change

can be determined from the equation. In this the symbols

c' and y denote the specific heat of the substance in the

liquid and in the solid condition, not however, as already

observed, the specific heat at constant pressure, but the

specific heat in the case in which the pressure changes with

the temperature in the manner indicated by equation (13).

The mode of determining this kind of specific heat will

be described in the next chapter. Here we will merelyby way of example give the numeyical values in the case

of water. The specific heat at constant pressure, i.e. that

specific heat which is simply measured at atmospheric

pressure, has in the neighbourhood of 0° the value 1 for

watgr, andy according to Personf, the value 0"48 for ice,

* Report, Brit. Assoc, 1854, p, 57.

t Cojuptes Rendvs, Vol. zxx, p, S26.




The specific heat for the case here considered has on the

contrary for water and ice the values c'=0"945 and k'=OQM.

For / we may take Person's value 79. We thus obtain

£=0-945 -0-631 +^3

= 0-314 + 0-289

= 0-603.

It is known that the freezing point of water can also

be lowered by protecting it from every sort of disturbance.

This lowering of temperature however only refers to the

commencement of freezing. As soon as this has begun, a

portion of the water freezes immediately such that the whole

mass of water is thereby warmed again up to 0°, and the re-

mainder of the freezing takes place at that temperature.

There is therefore no need to examine more closely the change

in the magnitude r' which corresponds to a lowering of the

temperature of this kind, and which is simply determined by

the difference of the specific heat of water and ice at constantpressure.

§ 6'. Passagefrom the solid to the gaseous condition.

Hitherto we have considered the passage from the liquid

to the gaseous and from the solid to the liquid condition.

It may however happen that a substance passes direct from

the solid to the gaseous condition. In this case three equa-

tions will hold of the same form as equations (15), (16)

and (17) of the last chapter, or (11), (12), (13) of this : we

must only remember to choose the specific heats and specific

volumes relating to the different states of aggregation, and

the quantities of heat consumed in the passage from one con-

dition to the other, in the manner corresponding to the pre-

sent case.

The circumstance that the heat expended is greater in

the passage from the solid condition to the gaseous than

from the liquid, leads to a conclusion which has already been

drawn by Kirchhoff *. For if we Consider a substance when

just at its melting point, vapour may be developed at this

temperature both from the liquid and from the solid. At

* Pogg. Ann., Vol. cm. p. 206.



FUSION AND VAPORIZATION OF SOLID BODIES. 173

temperatures above the melting point we have only to do

with vapour developed from a liquid, andat temperatures below

with vapour developed from a solid, leaving out of account

the special case mentioned in the last section, in which a

liquid kept perfectly Still remains fluid in spite of having

reached a lower temperature.

If for these two cases, i. e. for temperatures above and below

the melting point, we express the pressure of vapour p as a

function of temperature, and construct for each case the curve

which has the temperatures for abscissae and the pressures

for ordinates, the question arises how the curves correspond-

ing to the two cases are related to each other at the commonlimit, viz. the temperature of fusion. In the first place, so

far as concerns the value of p itself, we may consider it as

known by experience to be equal in the two cases ; and

thus the two curves will meet in one point at the tempera-

ture of fusion. But with regard to the differential coeflScient

-—,, the last of the above-named three equations shews that it

has different values in the two cases ; and thus the tangents

to the two curves at their point of intersection have different

directions.

Equation (17) of Chapter VI. which relates to the passage

from the liquid to the gaseous condition, may be written as

follows

dip _ Er . .

dT~ T{s-iT)^^''^•

To form the corresponding equation for the passage from

the solid to the gaseous condition, we should set on the left

hand the pressure of the vapour given off by the solid body,

which for distinction we may ca,ll P. On the right hand we

must put, instead of <t, which is the specific volume of"the

liquid, the specific volume of the solid which we may call t;

the difference thus indicated is however very small, since

these two specific volumes differ very slightly from each

other, and in addition are small in comparison with s, the

specific volume of. the substance as a gas. It is of more

importance to substitute for r, which is the heat required to

cause the passage from the liquid to the gaseous condition,

the quantity of heat required for the passage from the solid



174 OK THE MECHANICAL THEOat^ OF HEAT.

*to the gaseous condition. This latter equals r + r', where r'

is the heat required for melting. Thus in the present case

the equation is

dP_M{r + r') .-

dT,

T{s-t) ^''

Combining this eqjiation with (16), and neglecting the small

difference between <t and t, we have

dP dp Er' ,

dT dT~T{s-0-) ^ /

If we apply this equation to water, we must put T= 273,

r'= 79, s = 205, a- = 0-001, and giving^ the known value 424

we have

dP c?p_ 424x79 _dT drr 273x205"

If we wish to express the pressure in millimetres of mer-

cury, instead of kilogrammes per square metre, we must, as

remarked in Chapter VI. § 10, divide the above result by13'596; then putting for p and P the Greek letters tt and

IT, we have

^-^ = 0-044dT dT^^^-

It may be added for the sake of comparison that the diffe-

rential coefficient t™ has for 0° the value 0'33, according to

the pressures which Renault has observed at temperatures

just over 0°.



CHAPTER VIII.

ON HOMOGENEOUS BODIES.

§ 1. Changes of Condition without Change in the Con-

dition of Aggregation.

We will now return to the general equations of Chapter

V. and will apply them to cases, in which a body undergoeschanges which do not extend so far as to alter its condition

of aggregation, but in which all parts of the body are always

in the same condition. We will suppose these changes to be

produced by changes in the temperature and in the external

pressure. In consequence of these, changes take place in the

arrangement of the molecules of the body, which are indi-

cated by changes in form and volume.

With regard to the external force, the simplest case

is that in which an uniform normal pressure alone acts on

the body ; in this case no account need be taken of changes

in the body's form, in determining the external work, but only

of its alteration in volume. Here we may take the condition

of the body as known, if of the three magnitudes, tempera-

ture, pressure and volume, which we will denote as before

by T, p and v, any two are given. According as we choosefor this purpose v and T, or p and T, or v and p, so

we obtain one of the three systems of equations, which in

Chapter V. are numbered (26), (26) and (27) : these equa-

tions we will now use to determine the different specific heats

and other quantities, related to changes in temperature,

pressure, and volume.



176 ON THE MECHANICAL THEOHT OF HEAT.

§ 2. Improved Denotation for the Differential Coeffi-

cients.

If the above-named equations of Chapter V. are referred

to a unit of weight of the substance, the differential coefS-

cient -^ will denote in equations (25) the specific heat at

constant volume, and in equations (26) the specific heat at con-

stant pressure. Similarly -p has different values in (25)

and (27) and -^ has different values in (26) and (27). Such

indeterminate cases always occur where the -nature of the

question occasions the magnitudes chosen as independent

variables to be sometimes interchanged. If we have chosen

any two magnitudes as independent variables, it follows

that in differentiating according to one we must take the

other as constant. But if, whilst keeping the first of these

as one independent variable throughout, we then choose for

the other different magnitudes in succession, we naturally

arrive at a corresponding number of different significations

for the differential coefficients taken according to the first

variable.

This fact induced the author, in his paper " On various

convenient forms of the fundamental equations of the Me-

chanical Theory of Heat,"* to propose a system of denotation

which so far as he knows had riot been in use before. This

was to subjoin to the differential coefficient as an index the

magnitude which was taken as constant in differentiating.

For this purpose the differential coefficient was inclosed in

brackets and the index written close to it, a line being drawn

above' the latter, to distinguish it from other indices, which

might appear at the same place. The two differential co-

efficients named above, which represent the specific heat

at constant volume and at constant pressure, would thus be

written respectively (-T^jand \^ji)_- This method was

soon adopted by various writers, but the line was generally

* Beport of the Katuralists' Society of Zwrich, 1865, and Pogg. Ann., Vol.

cxxv. p. 353.



ON HOMOGENEOUS BODIES. 17T

left out for the sake of conveniencei More recently* theauthor introduced a simplei' form of writing, which yet

retained th? essential advantage of the method. This con-

sisted in placing the index next to d,

thesign of differentia-

tion. The brackets were thus rendered needless and also the

horizontal line, because no other index is in general placed

in this position. The two above-named differential coeffi-

cients would thus be written -^ and -~ ; and this methoddT dl

will be adopted in what follows.

§ 3. Relations between the Differential Coefficients ofPressure, Volume, and Temperature.-

If the condition of the body is determined by any two of

the magnitudes. Temperature, Volume, and Pressure, we

may consider each of these as a function of the two others,

and thus form the following six differential coeffiicients :

d,p d^p djo djV dJI d„T

dT' dv' dT' dp' dv ' dp'

In these the suffixes, which shew which magnitude is

to be taken as constant, may be omitted, provided we agree

once for all that in any differential coefficient that one of the

three magnitudes, T,p, v, which does not appear, is to be con-

sidered as constant for that occasion. We shall however

retain

themfor

the sakeof clearness, and because we shall

meet with other differential coefficients between the same

m.agnitudes, for which the constant magnitude is not the

same as here.

The investigations to be made by help of these six dif-

ferential coefficients will be facilitated, if the relations which

exist between t}iem are laid down beforehand. In the first

place it is clear that amongst the six there are three pairs

which are the reciprocals of each other. If we take v as

constant, T and p will then be so connected that each may

be treated as a simple function of the other. The same holds

with T and v where p is constant, and with v and p when T

* "On the principle of the Mean Brgal and its application to the mole-

cular motions of Gases." Proceedings of the Niederrhein. Ges. fur Natur-

und Seilkunde, 1874, p, 183.

c. 12



178 ON THE MECHANICAL THEOKY OP HEAT.

is constant. Hence we may put

_1 d^ »L =^ _L =^ (^^

<y dT' d^ dT' d^ df^ ^'

dp dv dv

To examine further the relation between these three

pairs, we will by way of example treat ^ as a function of

T and V. Then the complete differential equation for p is

Itp is constant, we must put in this equation,

d^dT

p= 6, dv =^dT;

whence it becomes

whence

= ^dT+^§§,dT:dT dv dT

^x^x^= -l (2).dv dT dp

By means of this equation combined with equations (1),

we may express each of the six differential coefficients by the

product or the quotient of two other differential coefficients.

§ 4. Complete Differential Equations for Q.

We will now return to the consideration of the heat taken

. in and given out by the body. If we denote the specific

heat at constant volume by C^, and at constant pressure by

Gp, and take the weight of the body as unity, we have

dT " dT~ "'

.We have also the equations (25) and (26) of Chapter V.,

which with our present notation will be written as follows

djQ^rpd^ dTQ__rpd£dv dT' dp dT'



ON HOMOGENEOUS BODIES, 179

Hence we can write down the following complete differen-

tial equations

dQ = C,dT+T^,dv (3),

dQ = C^dT-T^^^dp (4).

From these two we easily obtain a third differential equa-

tion for Q, which relates to t) and p as independent variables.

For multiplying the first equation by G^ and the second by

G„ subtracting, and dividing the result by (7^— 0„ we have

These three equations correspond exactly to those obtained

in Chapter II. for perfect gases, except that the latter are

simplified by applying the law of Mariotte and Gay-Lussac.

The equation expressing this law is

pv = RT,

whence we have

^ ^ -B, d^ _RdT V ' dT p'

Substituting these values in the above equations, and in the

last putting -^ for T, we get

dQ=G,dT+ ^dv,

dQ = G^dT-^dp,

CdQ = Q-_jQ pdv + Q Zc ^'^^'

These equations are the same as (11), (15) and (16) of

Chapter II.

The equations (3), (4) and (5) are not immediately inte-

grable, as has been already shewn with respect to the special

equations holding for gases. For equations (.3) and (4) this

12—2



180 ON THE MECHANICAL THEOKT OF HEAT.

follows from equations already given. If in the last equa-

tions of the systems (25) and (26) of Chapter :¥. we use the

symbols G, and G^,, and also the method above explained of

writing the differentialcoefficients,

wehave

dv~ dT" dp dr ^

^"

Whereas the conditions which must be fulfilled, if (3) and

(4) are to be integrable, are as follows

dflv^rpdlp d^, d^G^^ „djv dj)

dv dT^'^dT' dp dr dT'

By a similar but longer process we may shew that equa-

tion (5) is not integrable; as may at once be concluded from

the fact that it is derived from equations (3) and (4),

These three equations thus belong to that class of com-

plete differential equations which are described in the Intro-

duction, and which can only be integrated if a further relation

between the variables is given, and the path of the variation

thereby fixed.

§ 5. Specific Seat at Gonstant Volwne and at Gonstant

Pressure.

If in equation (4) we substitute for the indeterminate

differential dp the expression -^ dT, we introduce the special

case in which the body changes its temperature by dT, the

volume remaining constant. If we divide hj dTwe have on

the left-hand side the differential coefficient -^ , which is

the specific heat at constant volume and has been denoted

by (7„. Hence we obtain the following relation between

Cf. andC,:

C.= C,-2^g,x^ (^)-

Substituting in equation (5) the value of G^, — G„ given

by this equation, we obtain the following simpler form

dQ^G/^dv + G/^dp .(8).



ON HOMOGENEOUS BODIES. 181

If by means of equation (7) we proceed to determine the

specific heat at constant volume from that at constant pres-

sure, it is requisite first to make a slight change in the

equation. The differential coefficient -^ contained thereinaT

expresses the expansion of the body upon a rise of tempera-

ture, and may generally be taken as known, but the other

differential coefficient -^ cannot in general be determined

for solid and liquid bodies by direct experiment. However

from equation (2) we have

d^

d,p^_dTdT ' djv'

dp

In this fraction the numerator is tlie differential coeffi-

cient already discussed, and the denominator expresses, if

taken with a negative sign, the diminution of volume by an

increase of pressure, or the compressibility of the body ; and

this for a large number of liquids has been directly measured,

whilst for solids it may be approximately calculated from the

coefficients of elasticity. Equation (7) now becomes

(d^X

dp

If the specific heats are expressed not in mechanical but

in ordinary units, we may denote them by c„ and c,, ; the equa-

tion then takes the form

T\dT ,„,

''-'^Ê-W• ^^^)-

dp

In applying this equation to a numerical calculation we

iinust remember, :that in the differential coefficients the unit

of volume must be the cube of the unit of length which has



182 ON THE MKCHANICAL THEORY OF HEAT.

been used for determiningE : and that the unit of pressure

must be the pressure which a unit of weight exerts on a unit

of surface. If, as is usually the case, the coefficients of ex-

pansion and compression refer to other units, they must bereduced to those above mentioned.

Since the differential coefficifent ~- is always negative, the

specific heat at constant volume must always be less than

that at constant pressure. The other differential coefficient

jm is generally positive. In the case of water it is zero at

the temperature of maximum density, and accordingly the

two specific heats are equal at that temperature. At all

other temperatures, both above and below, the specific heat

at constant volume is less than that at constant pressure; for

although the differential coefficient -^ is negative below the

temperature ofmaximum density, yet, as it is the square of this

which occurs in the formula, it has no influence on its value.

As an example of the application of equation (7), we will

calculate the difference between the two specific heats in

the case of water at certain known temperatures. According

to the observations of Kopp (see his tables in Lehrbuch

der Phys. u. theor. Ghimie, p. 204), we have the following

coefficients of expansion in the case of water, its volume at

4° being taken as unity

at 0° -0^000061,

„ 25° +0-00025,

„ 50" +0-00045.

According to the observations of Grassi*, we have for the

compressilîlity of water the following numbers, which express

the diminution of volume upon an increase of pressure of one

atmosphere, in the form of a decimal of the volume at theoriginal pressure

at 0" 0-000050,

„ 25° 0-000046,

,i50° 0-000044.

* Ann. de Chim. et de Phys. 3rd ser. Vol. xxxi. p. 437, and Kronig'g

Joum. fiir Physik des Auslandes, Vol. ii. p. 129.




We will now, as an example, perform the calculation for

the temperature of 25°. The unit of length may be the

metre, the ' unit of weight the kilogram. We must then

take the cubic metre as the unit of volume ; and, since akilogram of water at 4° contains O'OOl cubic metres, we

d 1)

must, in order to obtain -^, multiply the coefficient of ex-

pansion given above by O'OOl. Thus we have

^ = 0-00000025 = 25 X lO"*.

For compression we must, by what has been said, take as

unit the volume which the water contained at the tem-

perature in question and at the original pressure (which latter

we may assume to be the ordinary pressure of one atmosphere).

This volume at 25° = 0-001003 cubic metres. Further we

have taken one atmosphere as unit of pressure, whereas we

must take the pressure of 1 kilogramme on 1 square metre;

in which case a pressure of one atmosphere is expressed by

10,333. Accordingly, we must put

dj,v _ 0-000046 X 0-001003 _ , . ^ „_i3^- Ipss^"""^^ •

Further, we have at 25°, 7= 273 + 25 = 298 ; and for Ewe will take Joule's value 424. Substituting these numbers

in equation (76), we get

298 25^ IT"

424 ^ 45 ^ 10"„ - c, = T^ X -7^ X ^7^5 = 0-0098.

Jn the same way we obtain. from the values given above

for the coefficients of expansion and compression at 0° and

50° the following numbers

at 0°, Cj,-c, = 0-0005,

„ 50°, c^-c, = 0-0358.

If we now give to Cp, or the specific heat at constant pres-

sure, the experimental values found by Regnault, we obtain

for the two specific heats the following pairs of values:






heat of water and ice, when the pressure changes with the

temperature in such a way that the temperature of melting

corresponding to the pressure at any moment is always equal

to the temperature which exists at that moment.

In the first case we have simply to give to -^ the value

corresponding to the intensity of pressure of the steam. Forthe temperature 100° this value is 370, taking as unit of

pressure a kUbgram per square metre. With regard to

^, the researches of Kopp give 0-00080 as the coefficient of

expansion of water at 100°, taking the volume of water at 4°

as unity. This number must be multiplied by Q-OOl, in order to

obtain the value of -^ in the case when a cubic metre is

taken as the unit of volume and a kilogram as the unit of

weight: we thus obtain the number 0'0000008. Lastly wewrite for T the absolute temperature for 100°, or 373, and

for E, as usual, 424. Then equation (9a) becomes

c = c^ -ll^

X 0-0000008 X 370 = c^ - 0-00026.

If we take for the specific heat of water at constant pres -

sure, and at 100°, the values derived from the empirical

formula of Regnault, we obtain for the two specific heats

which we wish to compare, the following simultaneous values

c, = 1-013,

c = 1-01274.

It thus appears that these two quantities are so nearly

equal, that it would have been useless to take account of the

difference between them in the calculations as to saturated

steam.

The consideration of the influence of pressure on the

freezing point of liquids shews that a great change in the

pressure only produces a very slight alteration in the freezing

point; hence in this case -^ must be very large. If we

assume, according ,to. the calculations in Chapter VII., that an




increase of pressure of one atmosphere lowers the freezing

point by O-OOTSS" C, we have

dp _ 10 333,

dT ~ 0'00733

h«nce equation (9ffl) becomes, giving to T the value at the

freezing point, viz. 273, and to E the value 424,

273 10333_^_ .908000^^

- «=* + 424"*000733 ^ dT~ °* + ^"^""" dT

Applying this equation first to water, we will talê Kopp'svalue for the coefficient of expansion of water at 0°, viz.

— 0000061 ; then, using the kilogram as unit of weight, and

the cubic metre as unit of volume, we have

^= - 0-000000061

whence, from the equation above,

c = c^- 0-055.

As Cj, is here = 1, being the ordinary heat unit, we have finally

c = 0-945.

Next, to apply the equation to ice, we will take the linear

coefficient of expansion of ice at 0'000051, following the

experiments of Schumacher, Pohrt, and Moritz ; whence thecubic coefficient will be 0-000153. In order to reduce this

number to the required units, we must multiply it by

0-001087, the volume of a kilogram of ice in cubic metres:

whence we obtain

^=0-000000166.dl

Substituting this value, the equation becomes

c = c,+ 0151.

According to Person* c, = 0-48 : hence we have finally

c = 0-631.

* Coinpte^ Ji^Ttdus, Vol. xzz. p. 526.




These values, 0'945 and 0'631, were those employed in Chap-

ter VII. for the calculation by which the relation between

the heat expended in fusion and the temperature of fusion

was determined.

§ 7. Isentropic Variations of a Body.

Instead of determining the kind of variation of condition,

which a body is to undergo, by means of an equation con-

taining one or more of the quantities T, p, v, we will now

lay down as a condition, that no heat is imparted to or with-

drawn from the body during its variation. This is expressed

mathematically by the equation

If this equation holds, we have further

that is, the entropy S of the body remains constant. Wewill therefore give to this kind of variation the designation

isentropic, already applied to the curves of pressure which

correspond to it : and will characterize the differential co-

efficients formed in discussing it by the index 8.

If in equation (3) we put dQ = 0, we have

If we divide this equation by dv, the differential coeffi-

dTcient -7-, thus obtained, refers to the case of an isentropic

variation, and hence we must write :

d,^=.^î£no)

dv c'^dT''^"^•

Similarly we obtain from equation (4),

x^, ....(11).d,T T __djj

dp Cr, dT

Applying either equation (5) or equation (7), we have

0-c/£dv^C.pp;




, d,v C, dTwhence -7— = — 7=r T~ •

dp Gp d^

dT

Applj'ipg equations (1) (2), this equation becomes

dp _0^ d£,^ ,>

dp'o^'^dp ^

^'•

If here we give to 0, its value from (7a), we obtain

d„v _d,v T /d^v\' , .

-d^—d^^-oXdr)^^^>-

If we take the reciprocal of (12), we obtain the equation

^^ =g.x^ (14).dv C, dv

This equation, if transformed in the same way as (12),

gives

dv dv C,\dTj ^''

These differential coeflScients between volume and pres-

sure, for the case of the entropy being constant, have been

applied to calculate the velocity of propagation of sound in

gases and liquids, as has been already described in Chap-ter II. for the case of perfect gases.

*

§ 8. Special Forms of the Fimdamental Equations for

an Expanded Bod.

Hitherto we have always considered the external force to

be a uniform surface pressure. We will now give an ex-

ample of a different kind of force, and will take the case of

an elastic rod or bar, which is extended lengthwise by a

tensional strain, e.g. a hanging weight, whilst no forces act

Upon it in a ti'ansVerse direction. Instead of a tensional we

may take a compressive strain, so long as the rod is not thereby

bent. This we should simply treat in the formulae as a

negative tension. The condition that no transverse force

acts on the rod would be exactly fulfilled only if the rod were




placed in vacuo and thus freed from the atmospheric pres-

sure. But, since the longitudinal strain, which acts on the

cross section, is very large in comparison with the atmos-

pheric pressure upon an equal area, the latter may be

neglected.

Let P be the force, and I the length of the rod, whenacted on by the force and at temperature T. The length,

and in general the whole condition, of the rod is under these

conditions determined by the quantitiesP and T; and we maytherefore choose these as independent variables.

Let us now suppose that by an indefinitely small change

in the force or temperature or both, the length I is increasedby dl. The work Fdl will then have been done by the

force P. Since however in our formulae we have taken as

positive not the work done but the work destroyed by a

force, the equation for determining the external work must

be written

dW=-Pdl (16).

TakingZ as

afunction of

Pand T, we may write this equa-

tion as follows

dW=-F{gdF+§dT);

dW ^dl dW „ dlwhence _=-P_; _=-P_.

Differentiating the first of these equations according to Tand the second according to P, and observing that, since

dPP and T are independent variables, -jfjj

= 0, we have

A(W] p.d'l

dT\dP) dPdT'

dP\dT)~dT dTdP-

If we subtract the second of these from the first, and

substitute for the difference on the left-hand side of the

resulting equation the symbol already employed for the

same purpose, viz. Df^, we have

I>rT = §,-- (17).




This value of 2)pj, we will apply to equations (12), (13),

(14), (15) of Chapter V., substituting P for x in this parti-

cular case throughout. We then obtain the fundamental

equations, in the following form :

dT\dP) dP\dT) dT ^ ^'

±(dQ\_±(dQ\_l_dQdT\dP) dP\dT)~ T dT ^ ''

g=^§(^«)'

dP [dTj~-^ dT' ^ '

§ 9. Alteration of Temperature during the extension of

the Mod.

The form of equation.(20)

indicates a special relation

between two processes, viz. the alteration in temperature

produced by an alteration in length, and the alteration

in length produced by an alteration in temperature. Thus

if, as is usually the case, the rod lengthens when heated

under a constant strain, and -^ is therefore positive, the

equation shews that t= is also positive;

whence it follows

that, if the rod is lengthened by an increase in the external

force, it must take in heat from without if it is to keep its

temperature constant, or in other words, if no heat is im-

parted to it, it will cool during extension. On the other

hand if, as may happen in exceptional cases, the rod shortens

when heated at constant pressure, and therefore -^^ is nega-

tive, then the equation shews that -j^ is also negative. In

this case the rod must give out heat, when lengthened byan increase of strain, if it is to preserve a constant tempera-

ture ;and if no giving out of heat takes place it must grow

warmer in lengthening.




The magnitude of the alteration of temperature which

takes place if the force is varied, without any heat being

imparted to or taken from the rod, is easily determined if

we form the complete differential equation of the first order

for Q, in the same way as we have already done in the case

of bodies under a uniform surface pressure. The differen-

f? (1

tial coefficient -jn is determined by equation (20), in which

we will write for -ttt, the fuller form -^ . In order to ex-dl dl

press the other differential coefficient -^ in a convenient^ dTform, we may denote the specific heat of the rod under

constant strain by C>, and the weight of the rod by M. Then

we have

dT-^^^^'

and the complete differential equation is as follows :

dQ =MG^dT+T^ (22).

If we now assume that no heat is imparted to or taken

from the rod, we must put dQ = 0, which gives

= MC^dT+T^dP.

dTIf we divide this equation by dP, the quotient -^p ex-

presses that differential coefficient of T according to P, in

the formation of which the entropy is taken as constant;

it should therefore be written more fully-f^-

We thus

obtain the following equation

d,T_ T dj'dF~~Mcr,''dT

^^^^

This equation was first developed, though in a slightly

different form, by Sir William Thomson, and its correctness

was experimentally verified by Joule*. The agreement of

• Phil. Traiw., 1869.



192 ON THE. MECHANICAL THEORY OF HEAT.

observation and tteory was specially brought out by a pbe-

nomenon occurring with India rubber, which had already

been noticed by Gough, but was then observed also by Joule

and verified by accurate measurements. So long as India

rubber is not extended at all, or only by a very small force,

it behaves, with regard to alterations in length produced by

alterations in temperature, in the same way as other bodies

i. e. it lengthens when heated and shortens when cooled. Whenhowever it is extended by a greater force its behaviour is

the opposite; i.e. it shortens when heated and lengthens

when cooled. The differential coefficient -^ is thu^ positive

in the first case and negative in the second. In accordance

with this it exhibits the peculiarity that it is cooled by an

increase of the strain, so long as the strain remains small,

but is heated by an increase of the strain when the strain

is large. This agrees with equation (23), according to which

j^- must always have the opposite sign to -~^

§ 10. Further Deductionsfrom the Equations,

The complete differential equation (22) may also be so

formed as to present T and I, or I and P, as the independent

variables. For this purpose we must first state the relation

which, holds between the differential coefficients of the

quantities T, I, and P. This relation will be expressed by

an equation of the same form as (2), viz,

' U/^Jr (tpt a^ J. _ , .

W''df''~dP^~^^^*''-

First, to -form the complete differential equation which

contains T and I as independent variables, we must consider

P as a function of T and I, and accordingly write (22) in,the

form

dQ = MG,dT+ T^(g dT+^ dl)

-[MO.^Tf/^)dT + T%'-§dl



ON aoMOGENEOTD'S! BODIESi 193^

Transforming the last term by means of equation (24),

we have

dQ=^(MG, +T^x^^)dT-T^,dl (25).

If we denote by 0, the specific heat at constant length,

the coefficient of dT in this equation must be equal to MCi;

whence

• ^'=^'+J4^f »Transforming this

by meansof

(24), we have

T UtI^'-^'-M^'-dj; (27)-

dp

Equation (25) assumes then the following simplified

form:

dQ = MO,dT-T^dl (28).

Secondly, to form the complete differential equation which

contains I and P as independent variables, we must consider

T as a function of I and P. Equation (22) then becomes

dQ =MC^{^+%dP) + T'^dP

= Mc/-^dl + (Mc/^^T'f^dP.

Transforming the coefficient of dP, we have

dQ = M-G/-^dl,-(MO^+T'^.^^)l^dP..

By equation (26) MOi can be substituted^for the expression

in brackets. The equation then becomes

dQ=MG,^dl + MC,^dP (29).

We will again apply equations (28) and (29) 'fo the case

c. 13



aad the secoead

194 ON THE MKCHANICAL THEORY OF HEAT.

of the rod when it neither receives nor gives out any heat,

and therefore dQ = 0. The first equation then becomes

'-i-mî?^''^•'

dl

But byequation

(24) we maywrite the latter thus

ds^ _ C^, dji ,„ w V

dp-o,''dP^^^^'

Giving to (7,' its value according to (27), we have

dl_dl T_ /dpZy , „.

dP~dP MOMt)"

^ ''

The relation between length and stretching force which

is expressed by the diflferential coefficient ^4,, as here de-

termined, is that which has to be applied to calculate the

velocity of sound in an elastic rod, in place of the relation

Expressed by the differential coefficient -5^, which is commonly

used, and which is determined by the coefficient of elasticity.

In the same way, to calculate the velocity of sound ii;i

gaseous and liquid bodies, we must use the relation between

volume and pressure expressed by -~ in place of that ex-

pressed by-J-

. We may however remark that in treating

of the propagation of sound, in cases where the force P is

not large, we may in equation (82), which serves to deter-

mine-J-,

substitute for the specific heat at constant tension,

denoted by Cp, the specific heat at constant pressure, as

measured in the ordinary way under the pressure of the

atmosphere.



CHAPTER IX.

DETERMINATION OF ENEBQT AND ENTBOPY.

§ 1. General Equations.

In former chapters we have repeatedly spoken of the

Energy and Entropy of a body as being two magnitudes of

great importance in the Science of Heat, which are determined

by the condition of the body at the moment, without its

being necessary to know the way in which the body has

come into this condition. Knowing these magnitudes, wecan easily make by their aid various calculations relating

to the body's changes in condition, and the quantity of heat

thereby brought into action. One of these, the Energy, has

already been made the subject of many valuable researches,

especially by Kirchhoff*, and the method of determining it is

therefore more accurately known. We will here treat of

Energy and Entropy simultaneously, and set forth side by

side the equations which serve to determine them.

In Chapters I. and III. the two following fundamental

equations, denoted by (III.) and (VI.) were developed

dQ = dU+dW. (Ill),

dQ^TdS. (YI).

Here U and S denote the Energy and Entropy of the

body, and dU and dS' the changes produced in them by an

indefinitely small change in the body's condition : dQ is the

quantity of heat taken in by the body during its change;

* Pogg. Am,., Vol. cm. p. 177.

13—2




dW the external work performed; and T tlie absolute tem-

perature at which the change takes place. The first equa-

tion is applicable to any indefinitely small change of con-

dition, in whatever way it takes place, but the latter can

be applied only to such changes as are in their nature

reversible. These two equations we will now write in

the form

dU=dQ-dW (1),

dS =^ (2).

Their integration will then determine U and 8.

Here we must first notice a point which has already been

mentioned with regard to energy in Chapter I., § 8. It is

not possible to determine the whole energy of a body, but

only the increase which the energy has received, whilst the

body was passing into its present condition from some other

which we choose as its initial condition ; and the same is also

true of the Entropy.

Now to apply equation (1). Let us suppose that the body-

has been brought into its present condition from the given

initial condition, the energy of which we will denote by U„,

by any convenient path, and in any way reversible or not

reversible ; and let us suppose dU to he integrated through

the range of this change in condition. The value of this

integral will be simply IT— t/,. The integrals oi dQ and

dW represent the whole quantity of heat which the bodyhas taken in, and the whole external work which it has per-

formed, during the change in condition. These we will de-

note by Q and W. Then we have the equation

U=U,+ Q-W... (3).

Hence it follows that if for any mode of passing from

a given initial condition to the present condition of the body,

we can determine the heat taken in and the work performed,

, we thereby know also the energy of the body, except as

regards one constant depending on the initial condition.

Next to apply equation (2). Let us suppose that the body

has been brought into its present condition from the given

initial condition, the entropy of which we denote by S„, by

any path whatever, but by a process which is reversible;



DETERMINATION OP ENERGY AND ENTROPY. 197

and let us suppose the equation integrated for this change in

condition. The integral of dS will have the value 8 — 8„:

whence we have

S-S, +j^ (4).

Hence it follows that if for any passage -of the body, by

a reversible method but by any path whatever, from a given

initial condition to its present condition, we can determine

-fp- 1 we shall thereby know the value of the entropy, ex-

cept as regards one constant depending on the initial con-dition.

/

§ 2. Differential Equations for ike Case in, which only

Reversible Changes take place, a/nd. in which the condition of

the Body is determined by two Independmi Variables.

If we apply both the equations (III.) and (VI.) to one

and the same indefinitely small and reversible change in the

body's condition, the element dQ will be the same in both

equations, and may therefore be eliminated. Hence we

have

^dS=dU+dW (5).

We will now assume that the condition of the body is

determined by two variables, wMch, as in Chapter VI., we

will generally denote by x and y, signifying by these certain

magnitudes to be fixed later on, such as temperature,

volume, pressure. If the condition is determined by x and y,

then all magnitudes, the values of which are fixed by the con-

dition of the body at the moment, without its being neces-

sary to know the way in which the body has come into that

condition, are capable of being represented by functions of

these variables; in which functions the variables must be'

considered as independent of each other. Accordingly the

entropy 8 and the energy U must be looked upon as

functions of x and y. On the other hand, the external work

W, as we have repeatedly observed, holds a completely dif-

ferent position in this relation. It is true that the differ-!,

ential coefficients of W, so far as concerns reversible changes,

may be considered as known functions of x and y : W itself

however caimot be represented by such a function, $,nd can




only be determined, if we have given not only the initial

and final conditions of the body, but also the path by which

it has passed from one to the other.

If in equation(5)

we put

dU^-^da, + ~^dy,

,^^ dW.,dW.

that equation becomes

^dS ,, rpdS , [dU ,dW\. ^{dU dW\,

As this equation must hold for any values whatever of dx

and dy, it must hold for the cases alnongst others in whichone or other of these differentials is equal to zero. Hence

it divides into the two following equations

.(6).

rpdS^md^doe dx dx

^dS^dJl dW.

dy dy dy

From these equations either S or U may be eliminated by

a second differentiation. We will first take U, as this gives

rise to the simplest equation. For this purpose we must

differentiate the first of equations (6) according to y, and the

second according to x. We shall write the second differential

coefficients of 8 and U in the ordinary manner : but the differ-

dW dWential coefficients of -y- and -^— we will write as followsdx dy

-r- (^- land-^- f-i— ). This is with the same obiect as inay \dx/ dx \dy J.

''

Chapter V., viz. to shew that they are not second differential

coefficients of a function of x and y. Finally we may observe

that Tf the absolute temperature of the body, which in this



DETERMINATION OF ENERGY AND ENTROPY. 199

investigation we assume to be uniform throughout the body,

may also be considered as a function of a; and y. We thus obtain

dT^dS ^ d'S

^<PU d idW\

dy dfX dxdy dxdy dy\dx}'

dT^dS .^d'S ^d'U d fdWs

dx dy dydx dydx dx\dy)'

Subtracting the second of these equations from the first,

and remembering that

d'S d^S , dÛ dÛ, and

dxdy dydx' dxdy dydx'

we have

dT ^dS dT^^dS_d/dW\ d /dW\

dy dx dx dy dy\dxj dx\dy)'

The right-hand side of this equation we have named in ,

Chapter V., "the work, difference referred to xy" and

have denoted it by D^/, hence we may put

J._d fdW\ d /dW\ .^

"' ,dy\dxj dx\~d^)'''''

and the previous equation becomes

dT dS dT dS_dy^dx dx^dy~^^ ^

^•

This is the differential equation, derived from equation (5),

which serves to determine {8).

Secondly, to eliminate8 from equations (6), we write them

in the following form :

dS 1 dU IdWcQ~'T dx'^ T dx'

d,8_i^dU 1 dWdy- T dy '^T dy-




Differentiating the first of these equations according to yi^

.and the second according to cs, we have

d'S 1 d?U 1 dT dU,d n dW\

dxdy~ T dxdy T dy dx '^ dy \T dxl'

d'S 1 d^_± dT dJJ ±n^\dydx T^ dydx T^ dx ^1^^ dx\T dyj

'

Subtracting the second of these equations from the first,

putting all the terms containing U in the resulting equation

on the left-hand side, and multiplying the whole equation by

I", we have

^ X— -— X— =r F— f- —) -A(l ^)~dy dx dx dy [dy \T dx j dx \T dy )

_

We will adopt a special symbol for the right-hand side of

this equation, viz.,

"" [dy\T dx) dx\T dy}\ ^''

and we may point out that between D^ and A,^ there is the

following relation:

." _„ dTdW dTdW^- = ^^--^d^ +S^ (10).

Using this symbol, the above equation assumes the form

• dTdU_dTdUdy dx dx dy

=K (11).

This is the differential equation, derived from equation

(5), which serves to determine U.

§ 3. Introduction of the Temperature as one of the

Independent Variables.

The above equations take a specially simple form, if the

temperature 2^ is chosen as one of the independent variables.

If we put T=y,vfe have

— = 1 — =dy ' dx



DETERMINATION OF ENERGY AND. ENTROPY, 201

We thus obtain from (10) the following expression of the

dWrelation between A,,, and D^

A,,= TD,,-

dx

.(12).

Equations (8) and (11) also become

dx '

.(13);dU ^

The differential coefficients of the two functions 8 and Uwith regard to x are thus known. For their differential

coefficients with respect to T we will take the expressions

which follow directly from (2) and (1) on the assumption

that the condition of the body is determined by T and x, viz.

dS

dT~




which is equation (15) of Chapter V. For the second equa-

tion the condition is

dx\dT)~dx\dT)~ dT^'''

This equation can be easily shewn to depend on the last.

For by (12)

Differentiating this equation according to T, we have

dK,_ dB^^ d (dW\'df~-^ dT '^^^'~dT\dx)-

Now, remembering that

'^~dT\dx) dx\dTJ'

we may write this equation as follows

dA^^ dD,,_d^(dW\

dT dT dx\dTj'

On substituting this value of -~ in equation (17), we are

brought back to the form of equation (16).

We have now to determine 8 and U themselves, by

integrating equations (15). Let us suppose that the body

has been brought into its present condition, by any path

we please, from an initial condition for which the quantities

T, 8, X, TJ have the values T^, 8^, x„, U„ respectively : and

let this particular change of condition give the range of the

integration. As an example, let us suppose that the body

is first heated from the temperature T„ to the temperature

T, while the other variable keeps its initial value x^, and

then that this other variable changes its value from «„ to x,

while the temperature remains constant. Then we have

(18).



(19).

DETERMINATION OF ENERGY ANB ENTROPY. 203

In botli these equations the first integral on the right-hand

side is a simple function of T, whilst the second is a function

of T and x.

Let us now make the opposite assumption, viz. that thechange of x first takes place at the initial value of T, and

then the change of T at the final value of x. Then weobtain

In both these equations the first integral on the right-hand

side is a simple function of x, and the second of T and x.

By what has been said above, we may choose any other

path whatever, instead of that which we have taken as our

example, in which path the changes of T and x may be

transposed in any way, or may both take place at once

according to any law. We should naturally in each special

case choose that path, for which the data requisite to perform

the calculation are most accurately known.

§ 4. Bpedal case of the Differential equations on the

assumption that the only external force is a Uniform Surface

Pressure.

If we assume as the only External Force a UniformPressure normal to the surface, we must put

dW=pdv.

„ dW dv , dW dvHence t^— Pt- ^^^d —r- =Pj-.dx ^ dx dy " dy

The expressions for D^ and A^ then assume peculiar forms.

Those for D^ have been already considered in Chapter V.

We have first

_ _ d( ^\ _^f ^^

^^~d^V'&) di[Pd^)'

"*

~ Uy \T dx) . dx\T^ dy)




In the last of these equations we will put for th^ sake of

brevity

^ =

f,(20),

whereby it becomes

A — T^\ ^ { ^'''\ ^ ^ (^'^\

"\jly\ dx) dx \ dyl

_

Performing the differentiation in these equations, and

remembering that, ,

=, , , we have°

dxay dydx

^.'%î-t-% w.

If the temperature T be selected as one independentvariable, whilst the other remains x as before, the expressions

become

_dp du d^ d^

^''~ dT dx - dx^ dT ^'^''

^.'=^-®4-£-S<^*)^

or, restoring to ir its value '^

^^'-Ûy d^ dx dT) ^dx^'^^'''

The equations (15) then assume the following forms :

dbt"" dT'^^^\dT''dx dx'^dTr"'

^^''^'

.jy fdQ dv\ ^ rmfdir dv dir dv\ , ,„_.



DETEBMINATION OF ENERGY AND ENTEOPT.

or written in another form,

205

.(26a).

If we further choose for the second variable, as yet unde-

termined, the volume v, and thus put x = v, we have

,

dv ^ dv

Hence the preceding equations become

(4î')^.(27).

If the pressure p be chosen as the second independent

variable, so that x=p, we have

and the equations become

'^-VS'^^-pp'

^H'dT-^S)'H'TT-4)''

(28).

§ 5. Application of the foregoing Equations to Homo-

geneous Bodies, and in particular to Perfect Gases.

For Homogeneous Bodies, where the only external force

is a uniform pressure normal to the surface, it is usual, as

at the end of the last section, to choose for independent

variables two of the quantities T,v, p; and -j^ then takes

the simple significations which we have several times alluded

to. Thus if T and -v are the independent, variables, and if




the weight of the body be a unit of weight, -^ signifies the

specific heat at constant volume : or, if T and p. are the

independent variables, the specific heat at constant pressure.

Equations (27) and (28) become in these cases

dU=C,dT+(T^-p)dv.(29),

d8 =^dT-dv

(30).

^rpdp,

dU={c^-p§)dT-{T%.p'^)dp\

If we wish to apply these equations to a perfect gas, we

may use the following well known equation

pv=IiT.

Hence, if T and v be selected as independent variables,

dT~ v'

and equations (29) then become

d8=G,fÊ'^'

dU=C,dT

As in this case G, must be regarded as a constant, these

equations can at once be integrated, and give

.(31).

.(32).

8 = S,+ CJog^+Rlogl^

If we choose T and p as independent variables, we mayput

dv R , dv ETdT

= — and ^- =P dp P



DKTERMINATION OF ENERGY AND ENTROPY. 207

accordingly equations (30) become

'^'''.f-^fl(S3,.

Whence "we obtain by integration

8 = S. + C^lo4-Blogl\^3^^_

The integration of the general equations (29) and (30) can

of course only be accomplished if, in (29), p and G, are

known functions of T and v, or if, in (30), v and. C^ are

known functions of T and p.

§ 6. Application of the Equations to a Body composed

ofmatter in two Different States

ofAggregation.

As another special case we may select the state of things

treated of in Chapters VI. and VII., viz, the case in which

the body under consideration is partly in one state of aggre-

gation and partly in another, and when the change, which

the body may undergo at constant temperature, is such that

the magnitudes of the parts in the two different states of

aggregation are altered, with a corresponding change in the

volume, but no change in the pressure. In this case the

pressure jp depends only on the temperature ; and we may

therefore put -^ = 0, by which equations (25) and (26) are

transformed as follows

(35).

As in Chapters,VI. and VII., let us denote by ilf the weight

of the whole mass, and by m the weight of the part in the

second state of aggregation ; and let us take m in place of x



208 ON I'HE MECHANICAL THEORY OF HEAT.

for the second independent variable; then equation (6),

Chapter VI., becomes

dv _

dm'

for which, by equation (12), Chapter "VI., we may substitute

dv

dmJ,

dp

dT

Then the ahove equations become

<^S = -^^dT+jjdm,

''H§-''S)''*f-^y-> (36).

To integrate these equations we may take as a starting

point the condition that the whole mass M is in the first

state of aggregation, that its temperature is T^, and that its

pressure is the pressure corresponding to that tempera:ture.

The passage from this to its present condition (in which

the temperature is T, and in which the part m of the whole

mass is in the second, and the part Jf— m in the first state of

aggregation) may be supposed to take place in the following

way :—First let the mass, still remaining entirely in the first

state of aggregation, be heated from T„ to T, and let thepressure change at the same time, in such a way that it

is always the pressure corresponding to the temperature at

the moment : then let the part m pass at temperature T from

the first to the second state of aggregation. The integration

has to be performed according to these' two successive stages.

During the first stage dm = 0, and thus it is only the first

term on the right-hand side which has to be integrated.

Here -^ has the value MG, where C signifies the specific

heat of the body in its first state of aggregation, and for the

case in which the pressure changes during the heating in the

way described above. This kind of specific heat has been

already discussed several times, and the conclusions drawn

in Chap. VIII., § 6, shew that where the first state of aggre-



DETERMINATION OP ENERGY AND ENTROPY. 209

gation is the solid or liquid, and the second the gaseous,

it may safely be taken, for purposes of numerical calculation,

as equal to the specific heat at constant pressure. It is only

at very high temperatures, for which the vapour tension

increases very rapidly with the temperature, that the difference

between the specific heat and the specific heat at constant

pressures is important enough to be taken into account.

Further, during the first change the volume v has the value

Ma, where a is the specific volume of the substance in the

first state of aggregation. During the second stage dT= 0,

and therefore it is only the second term on the right-hand

side of equation (3a) which has to be integrated. This inte-

gration can be at once performed for both equations, since

the coefficient of dm is a constant with regard to m. The

resulting equations therefore are

S = S, + Mf^^dT+^,

U=U„ +M/:(--!?) dT + mp

.(37).

If in these equations we put m = or m = il/, we obtain

the entropy and energy for the two cases in which the mass

is either entirely in the first or entirely in the second state of

aggregation, under the temperature T, and under the pressure

corresponding to that temperature. For example, if the first

state is the liquid and the second the gaseous, then if we put

m = 0, the expressions Prelate to .the case of liquid under tem-

perature T, and under a pressure equal to the maximumvapour tension at that temperature ; or if we put m = M, they

relate to saturated vapour at temperature T.

§ 7. delations of the Eocpressions T>^^ and A^^.

In concludingthis chapter it is worth while to

refer againto the expressions D,

following meanings

_ d fdW\ d (dW\

'^~dy\dxl dx\dy)'

^ and A„, which by (7) and (9) have~the

A„, =r d^ (l d'W\ _ d_ /I dM\

dy\Tdx) dx\T^dy}_

14




These are both functions of x and y : but if to determine

the condition of the body we choose instead of x and y any

two other variables which we may call ^ and ?;, we may form

corresponding expressions D^^ and Aj, as follows

I>i.'dvKd^J d^\dr,)'

Hn--

mi d (l dW\ _d^(l dM\

_d^n\T d^J d^\T dr,

,(38).

These are of course functions of f and 17, as the former

were of x and y. But if we compare one of them, e.g. that

for D^^, with the corresponding expresssion for D^^, we find

that these are not simply two expressions for one and the

same magnitude referred to different variables, but are

actually two different magnitudes. For this reason D^ has

not been called simply the work difference, but the work

difference referred to xy, so that it may be distinguished

from Di,,, the work difference referred to ^t]. The same holds

true of Aj,j, and A|,.The relation which exists between D^^,. and Z)j, may be

found as follows. The differential coefficients which occur in

the expression for i)^, in (38), may be derived by firfjt forming

the differential coefficients according to x and y, and then

treating each of these as a function of ^ and rj. Thus wehave

om^^dW dx dW d^

d^ dx d^ dy d^'

dW^dW dx dW dy

dr] ' dx drj dy drj'

Differentiating the first of these equations according to 77

and the second according to ^, and again applying the sameartifice, we have

d (dW\_

dvKd^J-

d^ fdW\ dxdx d fdWX dx dydx\dx) d^ dr] dy\dx J d^ dr]

dW d^ d /dW\ dx dy

dx d^d/r] dx\dy)dr]d^

d_ fdW\dydy dW d'y

dy\dy) d^ dr] dy d^dr]'"



DETERMINATION OF ENERGY AND ENTROPY. 211

d_ /dW\

^

d_ (dW\ dxdx d^ /dF\ dm dy

dx \ djo J d^ drj dy\dx J d/q d^

dW d^x d fdrW\dxdy

dx d^dt) dx\dy ) d^ drf

d /dW\dydy dW d^y

dy\dy) d^ d/rj dy d^d/ij

If we subtract the second of these equations from the first,

all the terms on the right-hand side disappear except four,

which may be expressed as the product of two binomial terms

in the following equation

dv\d^J d^[dvJ~\d^^dv'

dx dy\

'dv^'d^J

±fdM\dy\dx)

dx \dy )\'

Here the expression on the left side is jDj,, and the expres-

sion in the square bracket is D^^ Hence we have finally

i'.-Cdx dy dx dy

d^ dr) drj d^,)l>. .(39).

Similarly we may obtain

fdx

Hi'

d^dr)

dx dy\ .

'd^'^t^)^"'-

.(39a).

If we substitute one new variable only, e.g. if we keep the

variable x, but replace y by 17, we must put a; = |^ in the two

dcs dss

last equations, whence -tz.= ^ and 3- = 0- The equations

then become

--^D and A -Â, .(40).

If we retain the original variables, but change their order

of sequence, the expressions simply take the opposite sign, as

is seen, at once on inspection of (7)'and (9). Hence

(41).,^ = - J?^, and A,^ = - A^^,.

14—2



CHAPTER X.

ON NON-REVEKSIBLE PE0CESSE3.

§ 1. Completion of the Mathematical Expression for the

second main Principle.

In the proof of the second main principle, and in the

investigations connected therewith, it was throughout assumed

that all the variations are such as to be reversible. We must

now consider how far the results are altered, when the

investigations embrace non-reversible processes.

Such processes occur in very different forms, although in

their substance they are nearly related to each other. Onecase of this kind has already been mentioned in Chapter I.,

viz., that in which the force under which a body changes its

condition, e.g. the force of expansion of a gas, does not

meet with a resistance equal to itself, and therefore does

not perform.

the whole amount of work which it mightperform during the change in condition. Other cases of the

kind are the generation of heat by friction and by the

resistance of the air, and also the generation of heat by a

galvanic current in overcoming the resistance of the wire.

Lastly the direct passage of heat from a hot to a cold body, by

conduction or radiation, falls into this class.

We will now return to the investigation by which it was

proved in Chapter IV. that in a reversible process the sum of

all the transformations must be equal to zero. For one kind of

transformation, viz. the passage of heat between bodies of

different temperatures, it was taken as a fundamental principle

depending on the nature of heat, that the passage fropi a

lower to a higher temperature, which represents negative

transformation, cannot take place without compensation. On



ON NON-REVERSIBLE PROCESSES. 213

this rested the proof that the sum of all the transformations

in a cyclical process could not be negative, because, if any

negative transformation remained over at the end, it could

always be reduced to the case of a passage from a lower to ahigher temperature. It was finally shewn that the sum of

the transformations could not be positive, because it would

then only be necessary to perform the process in a reverse

order, in order to make the sum a negative quantity.

Of this proof the first part, that which shews that the sumof the transformations cannot be negative, still holds without

alteration in cases where non-reversible transformations occur

in the process under consideration. But the argument which

shews that the sum cannot be positive is obviously inappli-

cable if the process is a non-reversible one. In fact a direct

consideration of the question shews that there may very-

well be a balance left over of positive transformations ; since in

many processes, e.g. the generation of heat by friction, and the

passage of heat by conduction from a hot to a cold body,

apositive transformation

alonetakes place, unaccompanied by

any other change.

Thus, instead of the former principle, that the sum of all

the transformations must be zero, we must lay down our prin-

ciple as follows, in order to include non-reversible variations :

The algebraic svmi of all the transformations which occur

in a cyclical process must always he positive, or in the limit

equal to zero. ''

We may give the name of uncompensated transformations

to such as at the end of a cyclical process remain over without

anything to balance them ; and we may then express our

principle more briefly as follows ;

Uncompensated transformations must always he positive.

In order to obtain the mathematical expression for this

extended principle we need only remember that the sum

of all the transformations in a cyclical process is given

'

7^ . Thus to express the general principle, we must

write in place of equation V. in Chapter III.,

.,-/<

/f-io... ,.(ix).




Equation (VI.), Chapter III., then becomes

dQ<TdS (X.)

§ 2. Magnitude of the Uncompensated Transformation.

In many cases the magnitude of the Uncompensated

Transformation is obtained directly from! the equivalence

value of the transformations, as determined by the method

of Chapter IV. If for example a quantity of heat Q is

generated by any process such as friction, and this is finally

imparted to a body of temperature T, the uncompensated

transformation thus produced has the valuej, . Again, if a

quantity of heat Q has passed by conduction from a body

of temperature 2\ to another of temperature T^, then the

uncompensated transformation is Q(nr — -^-j. If a body

has passed through a non-reversible cyclical process, and we

wish to determine the resulting uncompensated transforma-tion, which we may call N, we have, by the principles

explained in Chapter IV., the equation

N-../f (1)..

As however a cyclical process may be made up of several

individualchanges

of condition in a givenbody, some of

which may be reversible, others non-reversible, it is in manycases interesting to know how much any particular one of

the latter has contributed towards making up the whole

sum of uncompensated transformations. For this purpose

we may suppose that after the change of condition which

we wish to enquire into, the variable body is brought by any

reversible process into its former condition. By this means

we form a smallei: cyclical process, in which equation (1)

may be applied just as well as in the whole process. Thus

if we know the quantities of heat which the body has taken

in during this process, and the temperatures which appertain

to them, the negative integral — 1 -yp- gives the uncom-

pensated transformations which have taken place. But as



ON NON-EEVEESIBLE PROCESSES. 215

the return to the original condition, which has taken place

in a reversible manner, cEln have contributed nothing to

increase this sum, the expression above gives the uncom-

pensated transformation which was sought, and which wascaused by the given change in condition.

If we examine in this way all the parts of the whole

process which are non-reversible, and thereby find the values

of N^, N.^ etc., which must all be individually positive, then

the sum of these gives the magnitude N relating to the

whole cyclical process, without requiring ua to bring under

review those parts of it which are known to be reversible.

§ 3. Expansion of a Gas imaccompanied by External

Work.

It may be worth while to examine more closely those

changes of condition, mentioned in § 1, which take place

in a non-reversible manner because the resistances to

be overcome are less than the forces at work ; our ob-

ject being to determine the amount of heat taken induring the process. As however there are a great number

of different changes ,of this kind, which are produced in a

great number of ways, we must confine ourselves to a few

cases, which are either especially noteworthy on account of

their simplicity, or have some special interest on other

grounds.

The general equation for determining the quantity of

heat which a body takes in, whilst it undergoes any given

change of condition, reversible or non-reversible, is as

follows

Q=U,-U,+ W .....(2);

in which U^ and 'U^ are the energy in the initial and final

conditions, and W is the external work done during the

variation.

To determine the energy we can employ the equations of

Chapter IX. If the only external force is a uniform pres-

sure, and if the condition of the body is determined by its

tempei-ature and volume, then we may use equation (29), viz.

dU=0,dT+{T^P,-p\dv (3).



2lQ ON THE MECHAlJlCAI, THEORY OF HEAT.

This must be integrated for a passage in some reversible

manner from the initial to the final condition. If the

temperature is equal in the two conditions, as we shall,

assume in the examples which follow, then theintegration

may be performed at constant temperature, and the result

will be, if we denote the initial and final volumes by

t)j and v^,

^.-^."i:(4H* <*'i

whence equation (2) becomes

^=£;(^S'"^)'^^-^^ ^^>

As the first and simplest case we may take that in which a

gas expands without doing any external work. We may sup-

pose a quantity of the gas to be contained in a vessel and that

this vessel is put in connection with another in which is a va-

cuum, so that part of the gas can pass from one to the other

without meeting any external resistance. The quantity of

heat which the gas must in this case take in, in order to keep

its temperature unaltered, is determined by putting W=0 in

the last equation ; thus we have

«=£;(^S-^)'^" (^>-

If we make the special assumption that the gas is a

perfect one, and therefore that^w —RT, we have

dp _RdT~ v'

whence

dT~^ v~ R v~P'

whence (6) becomes

<3 = (7).

As alreiady mentioned, Gay-Lussac, Joule, and Eegnaulthave experimented on expansion apart from external work.

Joule annexed to his experiment^, described in Chapter II., bywhich he determined the heat. generated in the compression







ON NON-SEVERSIBLE PROCESSES.

The equation thus becomes

219

<^-f!^{''%-py^-^p^(^^-^:)-.(8).

If the gas is a perfect one, the integral on the right-hand

side, as shewn in the last section, will = 0, and the equation

takes the simpler form

Q=pA%-'"d- •(9),

which expresses that in this case the heat taken in is onlythat corresponding to the work required for overcoming the

external pressure of the air.

If the heat is to be measured according to the ordinary,

not the mechanical unit, we must divide the right-hand side

of (8) and (9) by the mechanical equivalent of heat, whence

we have

Q=^(v,-v,). .(9a).

This kind of expansion has also been experimented on by

Joule. Having as before compressed air to a high pressure

in a receiver, he allowed it to escape under atmospheric

pressure. In. order to bring the escaping air back to the

original temperature, he caused

it, after leaving the receiver, to

pass through a long coil of pipe,

as shewn in Eig. 20, which was

placed together with the receiver

in a water calorimeter. There

then remained in the air only a

small reduction of temperature,

which it shared in common

with the whole mass of the calo-

rimeter. The cooling of the

calorimeter gave the quantity of

heat given off to the air during Fig. 20.



220 OS THE MECHANICAL THEORY OF HEAT.

its- expansion. Applying equation (9a) to this quantity of

heat, Joule was able to use this experiment as a means of

calculating the mechanical equivalent of heat. The numbers

obtained from three series of experiments gave a mean value

of 438 (in English measures 798) ; a value which agrees closely

with the value 444 found by the compression of air, and does

not differ from the value 424, found by the friction of water,

more widely than can be explained by the causes of error

inherent in these experiments.

§ 5. Method of Experiment used hy Thomson and Joule.

The above-mentioned experiments of Joule, in which air

contained in a receiver was expanded either by espaping into

another receiver or into the atmosphere, shewed that the

conclusions drawn under the assumption that air is a perfect

gas are in close accordance with experience. If however wewished to know to what degree of approximation air or any

other gas obeys the laws of perfect gases, and what are the laws

of kny variations that may occur from, the conditions of a per-

fect gas, then the above mode of experiment is not sufficiently

accurate ; since the mass . of the gas is too small compared

with that of the vessels and other bodies which take part in

the variation of heat, and therefore the sources of error

derived from these have too great an influence on the result.

A very ingenious method of making more accurate experi-

ments was devised by W. Thomson, and the experiments werecarried out by him and Joule with great care and skill.

Let us imagine a pipe, through which is forced a

continuous current of gas. At one place in this let a porous

plug be inserted, which so impedes the passage of the gas,

that even when there is a considerable difference between the

pressure before and behind the plug, it is only a moderate

amount of gas, suitable for the experiment, which can pass

through in a unit of time. Thomson and Joule used as pluga quantity of cotton wool or waste silk, which, as shewnin Fig. 21, was compressed between two . pierced plates, ABand CD.

Let us now take two sections, EFand GH, one before andone behind the plug, but at such a distance that the unequal

motions which may occur in the neighbourhood of the plug



ON NON-EEVERSIBLE PROCESSES. 221

H

Pig. 21.

are not discernible, and there is only a uniformcurrent of gas to deal with. Then the wholeprocess of expansion, corresponding to the differ-

ence ofpressure before and behind the plug, takes

place in the small space between these two sec-

tions. If then the current of gas is kept uniform

for a considerable time, a state of steady motionis produced, in which all the fixed parts of the

apparatus keep their temperature unaltered,

and neither take in nor give off heat. Then if,

as was done by Thomson and Joule, we surround

this space with a non-conducting substance, sothat no heat can either pass into it from without

or vice versa, the gas can only give out or take

in the quantity of heat expended or generated in the process;

and thus, even where this quantity is very small, a difference of

temperature may exist sufficient to be easily noticed andaccurately measured.

§ 6. Development of the Equations relating to the above

method.

In order to determine theoretically the difference of

temperature in the above case, we will first form the general

equations determining the quantity of heat which the gas

must have taken in, if the temperature at the second section

is to have any required value. From this we can readily

find the temperature at which the heat imparted will benothing.

The separate parts of the process in the present case are

connected partly with consumption, partly with generation of

heat. Heat will be consumed in overcoming the frictional

resistance due to the passage through the porous plug; whilst

by the friction itself the same amount of heat will be

generated. At certain points in the passage heat is consumed

in increasing the velocity; whilst at other points heat is

generated as the velocity decreases. To determine the quantity

ofheat which on the whole must be imparted to the gas, we mayleave out of account the parts of the process which balance

each other ; since it is sufficient for our purpose to know what

is the work which remains over as external work done or con-

sumed, and at the same time the actual perngianent change in




the vis viva ofthe current. For this we need only consider the

work done at the entrance of the gas into the space between

the sections, i.e. at section EF, and also at the exit from that

space, i.e. at section GH; and similarly the velocities of the

current at those two sections.

With regard to the velocities, the difference between their

vis viva can readily be calculated. If however they are at

each section so small as they were in Thomson and Joule's

experiments, their vis viva may be altogether neglected. It

then remains only to determine the work done at the two

sections. The absolute values of these quantities ofwork may

be obtained as follows. Let us denote the pressure at sectionISF by

Pj, and suppose the density of the gas at this section to

be such that a unit-weight at this density has the volume v^.

Then the work done during the passage through the section of

a unit-weight of the gas equals p^v^. .Similarly the work done

at section GJS will be p.jv^, where p^ is the pressure and v^

the specific volume at that section. These two quantities

must however be affected with opposite signs. At section

OH, where the gas is escaping from the given space, the

external pressure has to be overcome, in which case the work

done must be taken as positive ; while in section SF, where

the gas is entering the space and thus moving in the same

direction as the external pressure, the work must be con-

sidered as negative. Thus the net external work per-

formed on the whole will be represented by the difference

We have now further to determine the quantity of heat,

which a unit-weight of the gas must take in while it passes

through the distance between the two sections; suppos-

ing the gas to have at the first section, where the pressure is

p^, the temperature T^, and at the second section, where the

pressure is p^, the temperature T^. For this purpose wemust use the equation which applies to the case in which a

unit-weight of the gas passes from a condition determined bythe magnitudes ^^ and T^ into that determined by the

magnitudes p^ and T^, and performs in so doing the work

Pi%~Pi"v ^® therefore recur to equation (2), in which the

symbol w, denoting the external work, must be replaced by

P-2^2 ~Pi''i >l^snce we have

Q=U,-U^+p,v^-p,v^ (10).



ON NON-REVERSIBLE PROCESSES. 223

Here we need only to determine U^— U^, for which

purpose we can again use one of the differential equations for

U set forth in the last chapter. In this case it is convenient

to choose the differential equation in which T and p are the

independent variables, i.e. equation (30) of Chapter IX.

In this equation we may put

J3 ^£^2'+

p^cii? =pdv = d{pv)-vdp.

It thus takes the following form

dU=C,dT-(T^-v)dp-d{pv) (11).

This equation must be integrated from the initial values

T^, p^, to the final values T^, p^. The integration of the last

term can be performed at once, and we may write :

U,-U,==j^C^dT- {t^-v) dp']^ -p,v,+p,v, ....(12).

Substituting this value of U^- U^ in equation (10), we

obtain

Q^jÔ,dT-{T^^-v)dp'j (13).

Here the expression under the integral sign is the differ-

ential of a function of T and p, since C^ satisfies equation (6)

of Chapter YIII.

c (?<, _ y d'v

d^~ dr-

And thus the quantity of heat Q is completely determined

by the initial and final values of T and p.

If we now introduce the condition corresponding to

Thomson and Joule's experiments, viz. that Q = 0, then the

difference between the initial and final temperatures is no

longer independent of the difference between the initial and

final pressures, but on the contrary the one can be found




from the other. If we suppose both these differences indefi-

nitely small, we may use instead of (13) the following

differential equation

If we here put dQ = 0, we obtain the equation which

expresses the relation between dT and dp, and which may be

thus written

%-ki'TT-') (-)

If the gas were a perfect gas, and therefore pv = It T, we

should have

dv M _ V

hence the above equation would become

dT

dp

Thus in this case an indefinitely small difference of

pressure produces no difference of temperature; and the

same must of course hold if the difference of pressure is

finite. Hence one and the same temperature must exist

before and behind the porous plug. If on the contrary some

difference of temperature is observed, it follows that the gas

does not satisfy the law of Mariotte and Gay-Lussac, and by

observing the values of these differences of temperature

imder various circumstances, definite conclusions may be

formed as to the mode in which the gas departs from that

law.

§ 7. Results of the Experiments, and Equations of

Elasticity for the gases, as deduced therefrom.

The experiments made by Thomson and Joule in 1854*

shewed that the temperatures before and behind the plug,

were never exactly equal, but exhibited a small difference,

which was proportional to the difference of pressure in each

* Phil. Trans., 1854, p. 321. . :



ON KON-EEVERSIBLE PROCESSES. ^ 225

case. With air at an initial temperature of about 15V losses

of temperature were observed, which, if the pressure were

measured in atmospheres, could be expressed by. the equation

With carbonic acid the losses of heat were somewhatgreater ; with an initial temperature of about 19° they satis-

fied the equatioii ...

The differential equations corresponding to these two equa-

tions are as follows

^=0-26 and ^ = 1-15 (15.)dp dp ^ '

In a later series of experiments, published in 1862*,

Thomson and Joule took special pains to ascertain how the

cooling effect varies when different initial temperatures are

chosen. For this purpose they caused the gas, before reach-

ing the porous plug, to pass through a long pipe surrounded

by water, the temperature of which could be kept at will to

anything up to boiling point. The result showed that the

cooling was less at high than at low temperatures, and in the

inverse ratio of the squares of the absolute temperatures.

For atmospheric air and carbonic acid they arrived at the

following complete formulae, in which a is the absolute

temperature of freezing point, and the unit of pressure is the

weight of a column of quicksilver 100 English inches high:

^=0-92^ and^ = 4-64(yj.

If one atmosphere is taken as unit of pressure, these

formulae become

©andf.l.a(-;)- C).dp \TJ dp

With hydrogen Thomson and Joule observed in their

later researches that a slight heating effect took place instead

of cooling. They have however deduced no exact formula

* Phil: Tram., 1862, .p. 579,

C. 13



226 ON THE MECHANICAL THEOBY OP HEAT,

for this gas, because tlie observations were not sufficiently

accurate.

dTIf in the two formulae for -j-

-.given in (16), we substi-

tute for the numerical factor a general symbol A, they

combine into one general formula, viz,

f=^©' (")•

Substituting in equation (14), we obtain

dv

r^,-t; = â(j) (18).

According to Thomson and Joule, this equation should

be employed for gases as actually existing, in place of the

equation referring to perfect gases,

T—-v = Q^ dT "^"'

if we wish to express the relation which exists between

the change of volume and. temperature when the pressure is

kept constant.

If G^ is taken as constant, equation (18) can be integrated

immediately. Now it is only for perfect gases that it has

really been proved that the specific heat C^ is independent

of the pressure; and similarly it is only for perfect gases

that the conclusion derived from Regnault's experiments

is strictly true, viz. that G^ is al'feo independent of the

temperature. If however a gas differs very slightly from

the condition of a perfect gas, 0^, will have values differing

very slightly from a constant, and these differences may be

taken as quantities of the same order. Since in addition

the whole term containing G^ is only another- small quantity

of the same order, the differences produced in the equa-

tion by the differences of G^ will be small quantities ofa higher order, and such may in what follows be neglected

thus we may take C?, as constant. Then multiplying the

dTequation by -^ , and integrating, we have



ON NON-REVERSIBLE PROCESSES, 227

«=pr-iâ,(jy (19),

where P is the constant of integration, which in the present

case may be considered as a function of the pressure p.

According to the law of Mariotte and Gay-Lussac

we should have

«=f2^-(20);

and it is therefore advantageous to give the function Pthe form.

P=:^ + 7r,

P

where ir represents another function of p which however

can only be very small. Equation (19) then becomes

v=^R^+^T-iAC^{^^J(21).

This equation Thomson and Joule further simplified

as fbllows. The mode in which the pressure and volume of

a gas depend on each other varies less from the law of

Mariotte according as the temperature is higher. Those

terms of the foregoing equation which express this varia-

tion must thus become smaller as the temperature rises.

The last term is the only one which actually fulfils this con-

dition ; the last but one, irT, does not fulfil it. Accordingly

this term should not appear in the equation, and putting

TT = 0, we obtain

. = i2|-iJC7,(jy .(22).

This is the equation which according to Thomson and

Joule must be used for gases actually existing, in place of

equation (20) which holds for perfect gases.

An exactly similar equation was previously deduced

by Rankine*, in order to represent the variations from

the law of Mariotte and Gay-Lussac, found by Eegnault

* Phil. Trans., 18.54, p, 336.

15—2




in the case of carbonic acid. This, equation in its simplest

form may be written „™ a ,an\^ pv = BT--^ (23),

in which a like M is constant.If we divide this equation by p, and in the last term, which

is yery small, replace the product pv by the yery nearly equal

product BT, and finally write jS for the constant „ , we obtain

which is anequation of the same form

as(22).

§ 8. On the Behaviour of Vapour during Eacpansion under

Various Circumstances.

As a further example of the different results- which

may be produced by expansion, we will consider the behaviour

of saturated vapour. We will assume two conditions : (1) that

the vapour expanding has to overcome a resistance equal to

its" whole force of expansion ; (2) that it escapes into theatmosphere, and thus has only to overcome the atmospheric

pressure. Under the last condition we may make a distinc-

tion according as the vapour is separate from liquid in the

vessel from which it escapes, or is in contact with liquid,

which continually replaces by fresh evaporation the vapour

which is lost. In all three cases we will determine the

quantity of heat, which must be given to or taken from

the vapour during expansion, in order that it may -continue

throughout at maximum density.

First then let us suppose a vessel to contain a unit-

weight of saturated vapour, and let this vapour expand, e. g.

by pushing a piston before it. In so doing let it exert upon

the piston the whole expansive force which it possesses at

each stage of its expansion. For this it is requisite only that

the piston should move so slowly that the vapour which

follows should always be able to equalize its expansive force

to that of the vapour which remains behind in the vessel.

The quantity of heat Q, which must be imparted to this

vapour, if it expands so far as that its temperature falls from

a given initial value T^ to a value T^, is simply found by the

equation.. g^r^j^dT,.,, (24).



ON NON-REVERSIBLE PROCESSES, 229

Here h, is the iiiagnitude introduced ia Chapter VI., and

named the Specific Heat of Saturated Vapour. If, as is the

case with jpaost vapours, h has a negative value, the foregoing

integral, inwhich the upper limit

is lessthan the lower,

represents a positive quantity.

In tlie case of "water, h is given by formula (31) of

Chapter VI., viz., 8003n = I'OU rp— •

Applying this formula it is easy to calculate the value of Q for

any two temperatures 7', and T^. For example let us assume

that the steam has an initial pressure of 5 or of 10 atmospheres,

and that it expands until its pressure has fallen to one atmo-

sphere; then by Regnault's tables we must put T^= a+152"2,

or = a -1- 180".3 respectively, and T^ = a + 100; we thus obtain

the values Q = 52'1 or = 74"9 units of heat respectively.

In the second case we suppose that a vessel contains a

unit-weight of saturated vapour apart from liquid, and at a

temperature T^ which is above the boiling point of the

liquid; and that an opening is made in the vessel, so thatthe vapour escapes into the atmosphere. Let us proceed to

a distance beyond the opening such that the pressure of

the vapour is there only equal to the atmospheric pressure.

To insure that the current, of vapour shall expand in the

proper manner, let the vessel be fitted at the opening with a

trumpet-shaped mouth KJPQM (Fig. 22.) This mouth is not

actually needed in order that the

equations which follow may hold,

but merely serves to facilitate the

conception. Let KLM be a surface

within this mouth, such that the

pressure of the vapour is there only

equal to atmospheric pressure, and

its . velocity . so small that its vis

viva may be neglected. Wewill

further assume that the heat gene-

rated by the friction of the vapour

against the edge of the .opening

and the surface of the mouth is

not dissipated, but again imparted

to the vapour.

Now to determine the quantity"




of heat wliich must be imparted to the vapour during ex-

pansion, if it is to remain throughout in the saturated con-

dition, we will again apply the general equation (2) ; which

gives, if in this case we denote the heatby

^',

Q'Û,-U,+ W. (25);

here U^ is the energy of the vapour in its initial condition

within the vessel, If^ the energy of the vapour in its final

condition at the surface KLM, and W the external work done

in overcoming the pressure of the atmosphere.

The energy of a unit-weight of saturated vapour at

temperature T is given by the value of U in equation (37)

of Chapter IX., if we there put m = M=l, It is

P" [dTj; and yjjFirst give to T the initial value T^, and let

be the values of^, -^ , and p corresponding to this tempera-

ture. Again let T have the final value T^, and let^^, (—],

and Pj be the corresponding values. Then subtracting these

two equations from each other we have

^^-^-/I;(^-^S)^^-^^^1--4-^^©

-Pi 1- A^.(S.

.(26).

The external work which results from the overcoming ofthe atmospheric pressure p^, during an expansion fromvolume s, to volume s^, is given by the equation



ON NON-EEVERSIBLE PEOCESSE'S. 231

We will give another form to this expression. If, as in

-Chapter VI., we put s = u + a; where cr is the specific.volume

of the liquid, the equation becomes

Substituting for tt the expression given in equation (13),

Chapter VI., we have

F=i>,

^m. <^\+pA'^.-'^^) (27).

Now substituting in (25) the value of U^~ U^ from (26),

iand of W from (27), we arrive at the equation

+PîF.-<^d (28).

Here the heat is expressed in mechanical units. Toexpress it in ordinary heat units the right-hand side must

be divided hy E. As before we wiU put ^ = c;^ = r. At

the same time, since o- is a small quantity and varies very

slightly, we will neglect the quantities -7^ and (ff* — o-^).

Thus we obtain

g=\^cdT+ r,-r,^--^^[p,-p,) (29).

AdT),

This equation is well adapted for the numerical calcula-

tion of Q, since the quantities which it contains have all

been determined experimentally for a considerable number

of liquids.For water we have according to Regnault.

0+1^=0-305;

whence'"''"

cdT+ r^- r. = - 0-305 (2; - T^j.

J x



232 ON tn^ MECHA]<ncAL THEOET OP HEAT.

The quantities in the last term of equation (29) are also

sufficiently known, so that the whole calculation is easy.

For example if we take the initial temperature at five or

ten atmospheres, we have Q' = 19'5 or = l7"0 units of heat

respectively.

Since Q is positive, it follows that in this case also heat

must be imparted to the vapour, not taken from it, if no

part of it is to be allowed to condense ; which condensation

might take pilace not only at the opening, but equally well

inside the vessel. The quantity o? vapour so condensed

would however be less than in the first case, because Q' is

less than Q.

It may easily happen that the above equations give a

larger quantity of heat for .an initial pressure of five than

of ten atmospheres. The reason is that at five atmospheres

the volume of the vapour is already very small ;- and tlie

diminution of volume, when the pressure is raised to ten

atmospheres, is so small that the corresponding increase of

work during the escape of the vapour is more than balanced

by the excess of the free heat in the vapour at 180"3° over

that in the vapour at 152'2°.

L Lastly let us take the third case,

in which the vessel contains liquid

as well as vapour. Let the vessel

ABCD (Fig. 23) be filled to the level

EF with liquid, and above this with

vapour. Let PQ be the openingof escape, fitted, as ijj the last case,

with the truriipet-shaped mouth

KPQM, to regulate the spreading

out of the current of vapour. Let

there be some source of heat which

keeps the liquid at a constant tem-

perature T^, so that it continually

gives off new vapour to replace that

which escapes, and thus the condi-

j,.. tions of the escape remain always

the same.

This last circumstance makes an important distinction

between this case and the foregoing. The pressure, which

the vapour newly given oif exerts' on that alrea,dy existing.



ON KON-BEVBRSIBLE PROCESSES. ' SSS

performs work during tlie escape of the vapour, wMch must

be brought into the calculation as negative external work.

Let OHJ be a sui-face at which the vapour which passes

through has still thesame

expansive force

p^teijipefature T^,

and specific volume s, which exist within the vessel, and at

which the new vapour is given off. Again let KLM be a

surface at which the vapour passing through has simply the

expansive force equal to the atmospheric pressure p^. At both

surfaces we shall assume the velocity to be so small that

its, vis viva may be neglected. In its. passage from one

surface to the other, the vapour must continually have just

that measure of heat given to it or taken from it, which is

necessary in order to keep it wholly in the gaseous condition,

and completely saturated, and alsoin order that at the

surface KLM it may have the temperature T^ corresponding

to the pressure p^ {i.e. the boiling temperature of the liquid),

and the specific volume s^ belonging to that temperature.

We have now to enquire how large this quantity of heat Q"

must be for each unit-weight of the escaping vapour.

To determine this we may proceed as in the last case,

remembering that we have now a different value for the

external work. This value is the difference between the

work done at the surface GHJ, through which passes a

volume of vapour s^ at pressure p^, and that done at the

surface KLM, through which passes a volume s^ at pressure

p^. It is thus given by the equation

'W=p.,s^-p,s^.

Putting once more

p

dT

we have

If we now form for Q" an equation of the same form as

(25), and in it substitute for t/^— E/j the expression given in

(26), and for TTthe expression given above, the main terms in




these two expressions cancel each other, and there re-

mains

Q"=Jl{(^-P%)^^+ P^ - P^ +P.'^,-P^<r,....i^l).

If we transform this equation so. that it refers not, to

mechanical but to ordinary units of heat, and neglect the

terms containing cr, we arrive at the simple equation

Q" = r'cdT+ r,-r, (32).

For water the equation takes the forni

Q" = - 0-305(7,-2;);

and if we calculate the' numerical values' of Q" for an initial

pressure of five or of ten atmospheres, we obtain

Q" = — 15"9 or = — 24f5 units of heat respectively.

Since the values of Q are negative, it follows that in this

case heat must be taken out of the vapour, not imparted to

it. If this withdrawal of heat does not take place to a

sufficient extent at any place under consideration, then the

steam is there hotter than 100° and therefore superheated.

Here it is of course assumed that nothing but steam passes

through the first surface GHJ, and thus that there are no

particles ' of liquid mechanically carried off by the steam, as

may happen during violent ebullition.






On this head we need only refer to two results given in

Chapter VI. In most of the recent writings on the steam-

engine, amongst others the excellent work of de Pambour,

the foundation of the theory has been taken to be the

law of Watt, viz. that saturated steam when contained in a

non-conducting vessel remains during all changes of volume

steam of maximum density. In some, later writings, after the

publication of Eegnault's researches on the heat required to

evaporate water at different temperatures, the assumption is

made that steam partly condenses during compression, and

during expansion cools in a less degree than corresponds

to the reduction of density, and therefore passes into thesuperheated condition. On the other hand it is proved

in Chapter VI. that steam must behave in a way which

is different from the first assumption and the exact oppo-

site of the second assumption, viz. that it is superheated

during compression, and is partly condensed during expan-

sion.

Further it is assumed in the above writings, in default of

more accurate means of determining the volume of a unit-

weight of steam at different temperatures, that steam even at

its maximum density still follows the law of Mariotte and

Gay-Lussac. On the other hand it is shewn in Chapter VI.

that it departs widely from that law.

These two points have naturally an important influence

on the quantity of steam which passes from the boiler into the

cylinder at each stroke, and on the behaviour of this steamduring expansion. It is thus obvious that they are them-

selves sufficient to make it necessary that we should calculate

in a different way from that hitherto adopted the amount

of work which a given quantity of steam performs in the

steam-engine.

§2. On the Action

ofthe Stea/m-Engine.

In order to illustrate more clearly the series of processes

which make up the action of a condensing steam-engine,

and to bring out clearly the fact that they form a cyclical

process, continually repeating itself in the same manner,

the imaginary diagram (Fig. 24) may be employed, A is

the boiler, the contents of which are kept uniformly at



-APPLICATION TO THE STEAM-ÊNGINE, 237

a constant temperature

T, by means of a source of

heat. From this boiler a

part of the steam passes

into the cylinder B, and

drives the piston a cer-

tain distance upwards.

Then the cylinder is shut

off from the boiler, and

the enclosed steam drives

the piston still higher by

expansion. The cylinderis now put in connection

with the vessel C, which

represents the condenser.

It will be supposed that*

this condenser iskeptcold,Pig. 24.

not by injected water, but by cooling from without : this makesno great difference in the results, but simplifies the treaitment.

The constant temperature of the condenser we may call T„.

During the connection of the cylinder with the condenser, the

piston returns through the whole distance it has previously

traversed ; and thereby all the steam which has not of itself

passed into the condenser is driven into it, and there con-

denses into water. It remains, in order to complete the

cycle of operations, that this condensed water should be

brought back again into the boiler. This is effected by thesmall pump D, the working of which is so regulated, that

during the upward stroke of its piston it draws out of the

condenser exactly as much water as has been brought into it

by the condensation during the last stroke ; and this water is

then, by the downward stroke of its piston, forced back into

the boiler. When it has here been heated once more to the

temperature y^' all is again restored to its initial condition,

and the same series of processes may begin anew. Thus wehave here to deal with a complete cyclical process.

In the common steam-engine the steam passes into the

cylinder not at one end only, but at both ends alternately,

The only difference thereby produced is, however, that during

one up and down stroke of the piston two cyclical processes

take place instead of one, and it is sufficient in this case to




determine the work done during one process, in order to be

able to deduce the whole work done during any given time.

In the case of an engine without a condenser, we have only

to assume that it is fed with water at 100", and we may

then suppose it replaced by an engine with a condenser, the

temperature of the condenser being 100°,

§ 3. Assmnptions for the purpose of Simplification.

For the purpose of this investigation we will assume, as

has usually been done, that the cylinder is a non-conducting

vessel, and so neglect the exchange of heat which takes place

during each stroke between the walls of the cylinder and the

steam.

The vapour within the cylinder can never be anything but

steam of maximum density with a certain admixture of

water. For it is evident from the conclusions of Chapter VI.,

that during the expansion which takes place in the cylinder

after it is shut off from the boiler the steam cannot pass into the

superheated condition, because no heat is imparted to it from

without ; but must rather partially condense. It is true that

there are certain other processes, to be mentioned later, which

tend to produce a slight super-heating ; but this is prevented

from taking place by the fact, that the steam always carries

with it into the cylinder a certain amount of water in the

form of spray, with which it remains in contact. The exact

amount of this water is of no importance ; and since for the

most part it is diffused through the steam in fine drops, andtherefore readily participates in the changes of temperature

which the steam undergoes during expansion, no important

error will be introduced if at each moment under consideration

we assume that the tenaperature of the whole mass of vapour

in the cylinder is the same.

Further, to avoid too great complexity in the formulae, we

will first determine the whole work done by the steam pressure,

without examining how much of this is actually useful work,

and how much is expended on the engine itself in overcoming

friction, and in actuating the pumps required, besides the one

shewn in the figure, for the proper working of the machine.

This latter part of the work may be subsequently determined

and deducted from the whole, in the manner shown later oh;

It may further be remarkedwith regard to the friction between



APPLICATION TO THE STEAM-ENGINE. 239

the piston and cylinder, that the work expended upon this is

not to be regarded as wholly lost. For since heat is generated

by this friction, the inside of the cylinder is thereby kept

hotter than it otherwisewould

be,

andthe

powerof

thesteam

increased accordingly.

Lastly, since it is desirable to understand the working of

the most perfect machine possible, before enquiring into

the influence of the various imperfections which occur in

practice, we will in this preliminary investigation make two

further assumptions, which may afterwards be withdrawn.

The first is that the inlet pipe from the boiler to the cylinder,

and the outlet pipe to the condenser or to the atmosphere,are so large, or else the speed of the engine so slow, that

the pressure within the end of the cylinder connected with

the boiler is always equal to the pressure in the boiler itself;

and similarly that the pressure within the other end is always

equal to that in the condenser, or to the atmospheric pressure

as the case may be. The second is that there are no clear-

ance or waste spaces sufficient to affect the result.

§ 4. Determination of the Work done during a single

stroke.

Under the conditions just enumerated the amount of

work done during the cyclical process corresponding to a

single stroke may be written down by help of the results

obtained in Chapter YI. without further calculation, andtheir sum comprised in a simple expression.

Let M be the whole quantity of vapour which passes from

the boiler to the cylinder during one stroke ; of this let the

part OTj be in the form of steam and the remainder M — m^

in the form of water. The space which this mass occupies will

be (by Ch. VI., § 1) m, «, + Mfr, where w, represents the value

of u corresponding to T^, while o- is taken as constant and

therefore has no suffix. The piston has therefore been raised

so far that this amount of space is left under it ; and since this

takes place at the pressure p^ corresponding to T„ the work

done during the first pr6cess, which we may call W^, is given

by the following equation

W, = mû^p, + McTp^ (!)•




Let the expansion whichi succeeds to this continue so far

that the temperature of the vapour enclosed in the cylinder

falls from the value T^ to a value T^. The work done during

this expansion, which we may call W^, is given directly by

equation (62) of Chapter VI., if we there take T^ for the

final temperature, and corresponding" values for the other

quantities involved. Thus

F, = m, (p,- u,p,) - m, {p, - u,p,) + MCiT,-T;) (2).

On the return stroke of the piston, which now begins,

the vapour, which at the end of the expansion occupiedthe space m^ u^ + Mix, is driven out of the cylinder into

the condenser, overcoming the constant resistance jp^ The

pegative work thus performed is therefore given by the

equation

F, = -»»,Mji)„-if<rp„ (3).

Now let the piston of the small pump rise until it leaves-

under it the space Ma; the pressure then continues to bethe pressure p^ of the condenser, and the work done is

W.^Map, .-.....(4).

Finally, during the descent of this piston, the pressure pj

of the boiler has to be overcome, and we have therefore the

negative work

W, = -M<Tp, (5).

Adding these five equations we have for the whole work

done during the cyclical process by the steam pressure, or in

other words by the heat, which work we may call W, the

following expression

W = m,p, - m,p, +MG {T, - T,) + mû, (p, -p,). . . (6).

From this equation we must eliininate m^. If for m^ wesubstitute the value given by equation (13) Chapter VI., viz.

then wijj only occurs in ibe; product m^p^; for which equation




(55) Chapter VI. gives, if we substitute therein p and C for r

aad c, the expression

T T

Substituting this expression we obtain an equation in which

all the quantities on the right-hand side are known, since

the massesM and m^, and the temperatures 2\, T^, and 1\ are

supposed to be known directly, and the quantities p, p, and

^ are assumed to be known as functions of temperature.

§ 5. Special Forms of the Eoopression found in the last

section.

If in equation (6) we put T^=T^, we obtain the work

done in the case where the machine works without expan-

sion, viz.

W' = mû,{p,-p„) (7).

If on the other hand we assume that the expansion con-

tinues until the steam has cooled by expanding from the

temperature of the boiler down to that of the condenser (an

assumption which cannot be realized in practice, but forms

the limiting case to which we may approach as near as

is practicable) we have only to put 1\ = T^ ; whence we

have

W' = m,p,-m,p,+ MG{T-T,) (8).

Eliminating m^p^ from this equation by means of tha

same equation (55) Chapter VI., in which we must also put

y, = T„, we have

W'=m,p,'^j,^^''+ Mc{t,-T,+ Tm'^^ (9).

§ 6. Imperfections in the Construction of the Steam-

Engiiie.

With all steam-engines as actually constructed the

expansion falls much below the maximum value given at the

end of the last section. If for example we take the tempera-

ture of the boiler at 150°, and that of the condenser at 50",

then if the temperature of the steam in the cylinder is to be

lowered by expansion to the temperature of the condensei-,

c. , IG




the steam must be expanded (by the table given in § 13

of Chapter VI.) to twenty-six times its original volume. In

practice, on account of the many evils attending too high an

expansion, steam is not expanded beyond three or four times

its volume in general, or ten times at the very utmost. Such

an expansion, with an initial temperature of 151°, is shewn

by the same tables to lower the temperature to 100°, or 75°

at the utmost, instead of to 50°.

Besides this imperfection, which has already been taken

account of in the above investigation and included in equation

(6), the steam-engine is subject to several others, two of

which have been already expressly excluded from considera-

tion. These are, first the fact that the pressure in one end

of the cylinder is less than that in the boiler, and in the

other greater than that in the condenser, and secondly the

presence of waste spaces. We must now extend our previous

investigations so as to include these further imperfections.

• The influence upon the work done of the difference

between the boiler and cylinder pressures has been investi-

gated most fully by Pambour in his work TMorie des

Machines d Vapeur. The author may therefore be allowed,

before himself entering on the subject, to reproduce the most

important of these investigations, only making some changes

in the notation and omitting the quantities which refer to

friction. It will thus be easier to shew how far Pambour's

results are no longer in accordance with our present know-

ledge on the subject of heat, and at the same time to connectwith it the new method of investigation, which in the

author's opinion must take its place.

§ 7. Pambour's Formulae for the relation between

Volume and Pressure.

Pambour's theory has its foundation in the two laws

already mentioned, which at that time were generally applied

to the case of steam. The first is the law of Watt, viz. that

the sum of free and latent heat is always constant. Fromthis law, as already mentioned, was drawn the conclusion that

if a certain quantity of steam at maximum density wereinclosed in a non-conducting vessel, and the contents of this

vessel then increased or diminished, the steam would neither

.be superheated nor partially condensed, but would remain




precisely at maximum density; and that this would take

place without any reference to the way in which the change

of volume was produced, or to whether the steam had to

overcome a resistance corresponding to its expansive force,

or not. This was the condition of things which Pambour

supposed to exist in the steam cylinder; and he also assumed

that no clfange of importance would be occasioned by the

presence of the particles of water which are always mixed

with the steam.

Next, to determine more closely the connection between

volume and temperature, or between volume and pressure,

in the case of steam at maximum density, Pambour applied

further the law of Mariotte and Gay-Lussac. If we take the

volume of a kilogram of steam at 100° and at maximumdensity to be 1*696 c. ipa., as given by Gay-Lussac, and if we

remember that the corresponding pressure of one atmo-

sphere represents a weight of 10333 kilograms to the square

metre ; and if for any other temperature T we call v the

volume and p the pressure ;

then the law of Mariotte andGay-Lussac leads to the following equation:

,.„- 10333 273 + r ,,„.

« = 1-696x-^x2-73^^:^0

(10).

Here we have only to substitute for p the value given by

the table of pressures, and we can then determine the volume

corresponding to any temperature, provided the foregoing

assumptions are correct.

Since in the formulae for the work done by the steam-

engine I pdv plays a principal part, it was necessary for the

convenient treatment of this integral, that the simplest

possible relation between p and v should be used. The

equations which would have been obtained, if it had been

sought to eliminate the temperature T from the foregoing

equation by means of one of the ordinary empirical fortnulae

for p, would have been too complicated : Pambour accordingly-

preferred to form a special empirical formula for the purpose,

and following the example of Navier he gave it the following

general form

"-u^--.("'

-16—2




wtereB and h are constants. These constants he endeavoured

to fix so that the volumes calculated by this formula might

agree as nearly as possible with those calculated by the

formula given above. As however sufficient accuracy cannotthus be arrived at in the case of all the pressures which are

met with in the steam-engine, he made use of two different

formulae in the cases of engines with and without condensers.

The first, for condensing engines, is as follows

20000 ,„ ,

"=1200+^ (l^'^)'

This agrees best with the formula (10) for pressures between

§ and 3^ atmospheres, but may be applied within somewhat

wider limits, say ^ and 5 atmospheres. The second, for non-

condensing engines, is as follows

21232,,,,,

^ = 3020+^- (11^>-

This is moSt accurate between 2 and 5 atmospheres, and may

be applied anywhere between 1 J and 10 atmospheres.

§ 8. Pambour's Determination of the Work done during

a single stroke.

The quantities needed for determining the work done, and

depending on the dimensions of the engine, will be here

denoted in a manner somewhat differing from that of Pambour.

The whole space within the cylinder, including the wastespace, which is left open for steam during a single stroke,

we shall call v'. The waste space we shall call ev' and the

space swept through by the piston (1 — e) v. That part of

the whole space, which is left open for the steam up to the

moment when the cylinder is shut off from the boiler, again

inclusive of the waste space, we shall call ev'. Then the

space swept through by the piston during the entrance of

steam will be denoted by (e — e) v, and that swept through

during expansion by (1 — e) v.

First to determine the work done during the entrance of

the steam. For this purpose we must know the actual

pressure in the cylinder at this time, which must be less than

that in the boiler, otherwise there would be no flow from one

into the other. The ampunt of this difference cannot how-




ever be fixed in general terms, because it depends not only on

the construction of the engine, but also on how widely the

valve in the steam pipe has been opened by the engineer,

andon the speed with which

theengine

is

moving. By achange in these circumstances the difference may be made to

vary within wide limits. Again the pressure in the cylinder

may not remain constant during the whole time the steam

is entering, because the speed of the piston, and also the

opening left by the valve, may be made to vary during this

time.

With reference to this latter point Pambour assumes that

the mean pressure, to be used for determining the work done,

may with sufficient accuracy be taken to be the same as the

final pressure which exists in the cylinder at the moment

when it is shut off from the boiler. The author does not

think it desirable to introduce into the general formulae an

assumption of this kind, although in the absence of more

exact data it may fairly be resorted to for the purpose of actual

calculations ; but he is bound to follow Pambour's method, in

order to complete the exposition of his theory.

The actual pressure at the moment when the steam is

shut off, Pambour determines by means of his equation,

as given above, between volume and pressure ; assuming that

special observations have beenmade to determine the quantity

of steam which passes from the boiler to the cylinder during

an unit of time, and therefore during each stroke. WewUl, as before, denote by M the whole quantity which passesinto the cylinder during one stroke, and by m the portion

which is in the condition of steam. As this quantity M, of

which Pambour only recognizes the part which is in the

condition of steam, fills at the moment when the cylinder is

closed the space ev', we have by equation (11),

e^==^ (12),

where p^ is the pressure in the cylinder at that moment.

Hence

P.-'^-i (12^).

If we multiply this equation by (e — e) ?;', which is the




space swept through by the piston up to this moment, we

have the following equation for the first part of the wort

done:

W.^mBx^-^-i/ie-e)!) ..,,(13).

The law of Tariation of the pressure during the exp£|,nsion

is also given by equation (11). If v is the volume and p the

pressure at any moment, then

mB ,

This expression we must substitute in fpdv, and then

integrate this from vêv to v = v'. Thence we obtain for

the second part of the work done

W^ = mB\og^-^v' 0.-e)l (14).

Next, to determine the negative work done by the resis-

tance during the return stroke, we must know the value ofthat resistance. Without entering at present into the ques-

tion how this resistance is related to the pressure in the

condenser, we will denote the mean pressure by p,; then the

work done will be given by.

Tf, = -«'(l-6)^„ (15).

Finally there remains the work which must be expended

in forcing back into the boiler the quantity of liquid M.Pambour has taken no special account of this work, but

included it with the friction of the engine. Since, however,

for the sake of completing the cycle of operations, it has been

included in the author's formulae, it will be investigated here

in order to facilitate the comparison. If p^ be the pressure

in the boiler, and p„ in the condenser, then equations (4) and

(5) show, asin the example already considered, that this

work is on the whole given by

W, = -M<r{p,-p,) (16).

For the present case, where p^ is not the pressure in the

condenser itself, but in the end of the cylinder which is open.to the condenser, this equation is not quite exact ; but since

on account of the smallness of a the value of the whole




expression is almost too small to be taken into account,

d fortiori we may neglect an error which is small even in

comparison with that value; and we shall therefore retain

the expression inthe form given above.Adding these four several quantities of work together, we

obtain the following expression for the whole work done

during the cyclical process

W = mB (iZL%iogl) _^'(l_e)(J, + p^) ^ Ma (p,- p,)

(17).

§ 9. Pamiour's Value for the Work done per Unit-weight

of Steam.

If instead of the work done in one stroke, during whichthe quantity of steam us^d is m, we prefer to find the workdone per unit-weight of steam, all that is needed is to divide

the foregoing value by m. Let us denote by I the fraction

M . . .

—, which gives the ratio of the whole mass which passes into

the cylinder to that part of it which is in the form of steam,

and is therefore somewhat greater than 1 ; by F the frac-

v' .

tion — , i.e. the space which on the whole is occupied by the

unit-weight of steam in the cylinder ; and by W the fraction

W

—, i,e.

the work done perunit-weight of steam.

Then wohave

W=B(^-^+ log'^yril-ê)ib+p,)-la(p,-p,)

(18).

In this equation there is only one term which involves

the volume F,, and this contains F as a factor. Since this

term is negative, it follows that the work, which can be

obtained from one unit-Tsreight of steam, is the greatest, other

things being equal, when the volume which that steam occu-

pies in the cylinder is the least possible. The least value of

this volume, to which we may continually approximate, but can

never exactly obtain, is that which is given by the assumption

that either the engine goes so slowly, or the steam-pipe is so







250 ON THE ]ilECHANICAI< THEORY OF HEAT,

Let the mass fi, which is not forced back into the boiler,

be first cooled in the liquid condition from T^ to T„, and then

at this temperature let the paxt /*„ be transformed into steam

the piston receding so far, that this steam can againoccupy

its original space.

Thus the mass M+ /j, will have passed through a complete

cyclical process, to which we may now apply the principle

that the sum of all the quantities of heat taken in during

a cyclical process must be equal to the total external

work done. In this case the following quantities of heat

have been taken in

(1) In the boiler, where the mass M has been heated

from T^ to T^, and at the latter temperature the part m^

transformed into steam, we have the quantity of heat

m,p, + MCiT,-T;),

(2) During the condensation of m^ at temperature T^, wehave

(3) During the cooling of the part /t from T^ to T^, wehave

(4) During the vaporization of the part fi^ at temperature

T^, we have

The whole quantity of heat taken in, which we maycall Q, is thus given by

The quantities of work are obtained as follows

(1) To determine the space swept through by the

piston during the entrance of the steam, we must remember

gaseous condition at tlie end as at the beginning, we need only assume that

the water forced back into the boiler is not only in quantity, but also in

its actual molecules, the same as that which left the boiler previously; andthat when this water takes up the temperature Tj, the quantity m, which wasformerly vapour, is again vaporized, whilst an equal quantity of the existing

steam is condensed. Por this purpose there is no need that the whole massin the boiler should take in or give out any heat, because that required for

the vaporization, and that generated by the condensation, exactly balanceeach other.



APPLICATION TO T5E STEAM-ENGINE. 251

that the whole space taken up hj M+ fi at the end of this

time is given by

From this we must subtract the waste space. Since thisis filled at the commencement by a mass fi at temperature T^,

of which the part fi^ is in the condition of steam, it maybe represented by

fi^% + fia-.

If we subtract this expression from the former, and multiply

the remainder by the mean pressure p\, we obtain for

the first part of the work done

(2)' The work done in the condensation of the mass

Wjis

(3) The work done in forcing the mass M back into the

boiler is

-Map,.

(4) The work done in vaporizing the part m„ is

Adding these four quantities together we obtain the

following expression for the whole work done

TF-m^ {P^-P.)-Ma (p,

-p,') -^

i^MPi'-Po)(20).

Since Q = W,'we may equate the expressions in (19) and

(20) ; and bringing to one side of the equation the terms

containing m^, we obtaiA

m,[p, + u,{p:-p,)]= m,p, + MGiT,-T,)+f.,p,-,iCiT-T,)

+ fiMP^-Po)+^^(Pi-Pi) (21)-

By means of this ecjuation the quantity m^ is given

in terms of quantities assumed to be already known.

§ 11. Divergence of the above Resvlts from Fambour's

Assumption.

In cases where the mean pressure p/ is much larger

than the final pressure ^j,—e.g. if we suppose that during

the greater part of the time of admission of the steam.




the pressure in the cylinder is nearly- the same as in

the boiler, and that it is only by the expansion that .the

pressure is finally brought down to the value p^,—it may

happen that the value found for mîs less

thanm^

+ji,^,

and accordingly that a part of the original steam must

have condensed. On the other hand if p^ is only a little

greater or even a little smaller than p.^, then the value

of wij, will be greater than m^ + fi^. This, latter is in general

true of the steam-engine, and in particular holds for the

case assumed by Pambour, in which p^ =p^.

Here as in Chapter VI. we have arrived at a result

widely diverging from the views of Pambour. Whereashe assumed for the two different kinds of expansion, which

succeed each other in the steam-engine, one and the same

law, according to which the original quantity of steam

can neither grow less nor greater, but must always remain

exactly at maximum density, we have been led to two

separate equation", which indicate an opposite condition

of things. In the first expansion, during the admission of

sbeam, there must by equation (21), be a continual genera-

tion of fresh steam ; in the further expansion, after the

steam is cut off, and while it is doing the full work

corresponding to its forceof expansion, there must by

equation (36) of Chapter VI. be a condensation of some part

of the existing steam.

Since these two opposite processes of increase and

diminution of steam, which must also exert an oppositeinfluence on the amount of work done by the engine,

partly go to cancel each other, the final result may in

certain circumstances be nearly the same as under the

simpler assumption of Pambour. We must not however

on this account cease to take into consideration the difference

we have discovered, especially if our object is to determine

what effect a change in the arrangement or speed of the

engine will have on the amount of work done.

§ 12. Determination of the'Work done during one stroke,

taking into consideration the Imperfections already noticed.

We may now return to the complete cyclical process

performed by the steam-engine, and treat its several parts

one after another in the same way as before.



APPLICATION TO THE STEAM-ENGINE, 253

The mass M flows from the boiler, in which the pressure

is p,, into the cylinder. Of this mass the part m^ is in

the condition of steam, the remainder of water. The meanpressure in the cylinder during this time we will as before

call p^, and the final pressure p^. The steam now expands,

until its pressure has fallen from p^ to p^, and its temperature

from jTjj to Ty Then the cylinder is opened to the condenser,

the pressure in which is p^, and the piston moves back

again through the whole length of the stroke. The back

pressure which it has to overcome is on account of its

comparatively rapid motion somewhat larger than^^, and the

mean value of this back pressure we will for the sake of dis-

tinction call p^. The steam which remains in the waste space

at the end of the return stroke, and must be taken into ac-

count for the next stroke, has a pressure which may not be

equal either to p^ or to p„', and therefore must be denoted

by />„". It may be greater or less than p^, according as the

steam is shut off from the condenser a little before or after

"the end of the return stroke ; for in the former case the steam

will be compressed still further, while in the latter it will

have time to escape partially into the condenser and so to

expand again. Finally the mass M must be returned from

the condenser into the boiler, during which as before the

pressure p„ assists the operation, and the presstire p^ has to

be overcome.

The quantities of work done during these processes will

be represented by expressions very similar to those in thesimpler case already considered : a few obvious changes must

be made in the suiBxes to the letters, and the quantities

added which refer to the waste space. We thus obtain the

following equations

During the admission of steam we have, as in § 10,

only writing «," instead of w„,

W, = {m,u^+Mcr- m,u;') pi (22).

During the expansion from pressure p^ to p^, we have as

in equation {^^) of Chapter VI., writing M-^yi, instead of M,

TF', = mûjp^ - m,uj)^ + 'm^Pi- m^P^ +{M+fi)C (T, - T^

(2S).





APPLICATION TO THE STEAM-ENGINE, 255

§ 13. Pressure of Steam in th6 Cylinder during the dif-

ferent Stages of the Process, and corresponding Simplifications

of the Equations.

To obtain numerical values from equations(27),

wemust first determine more accurately the quantities p^,

No general law can be laid down as to the mode in which

the pressure within the cylinder varies during the admission

of steam, because the opening and closing of the steam pipe

is performed with different engines in different ways. For

the same reason no fixed general value can be laid down for

the relation between the mean pressure p^' and the final

pressure p^ taking the latter in its strictest meaning. This

is however possible if we make a slight change in the signifi-

cation of ^2-

The shutting off of the cylinder from the boiler cannot

of course be instantaneous: the necessary motion of the

valve or cock must consume a certain time, less or greater

according to the arrangement of the gear. During this time

the steam in the cylinder expands somewhat, because as the

opening narrows the quantity of fresh steam which enters is

less than corresponds to the speed of the piston. We may

therefore assume in general that at the end of this time the

pressure is already somewhat less than p^.

Again, if we do not limit ourselves to taking the end of

the time required to close the valve as the moment of cut-off,

but allow ourselves some freedom in fixing this moment,other values may be obtained for p^. The moment may be

so chosen that, if at this moment the whole mass M had

already been admitted, a pressure would then exist exactly

equal to the mean pressure calculated for all the time up to

this instant. If we substitute for the actual cut-off the

instantaneous cut-off as thus determined, the resulting error

in the calculation of the work will be quite insignificant.

With this modification therefore we can fall in with Pamrhour's assumption, that

_Pi'= p„, but we must then for each

particular case make a special investigation of the circum-

stances, in order to determine accurately the moment to be

fixed for the cut-off.

As regards the back-pressure p^, which exists during the

return stroke, the difference Po'—Po is obviously smaller,




other things being equal, as p^ is smaller. It is therefore

smaller for a condensing engine than for a high-pressure

engine, where jo„ equals one atmosphere. In the case of the

most important high-pressure engine, the locomotive, there

is generally a special circumstance which tends to increase

this difference, viz. that the steam is not usually led into the

air by the shortest and widest channel possible, but is led

into the chimney, and there allowed to escape through a

somewhat contracted blast-pipe, in order to increase the

draught. In this case an accurate determination of this

difference is of importance to obtain a trustworthy result.

It must also be remembered that this difference is not con-stant even with the same engine, but depends upon the

speed, and the law of this dependence must therefore be

investigated. This subject, with the researches made upon

it, will not however be entered on here, since it has nothing

to do with the present application of the Mechanical Theory

of Heat.

In the case of engines where the exhaust steam is not

thus employed, and especially with condensing engines, p„'

differs so little from p^, and therefore varies so slightly with

the speed, that it is usually sufficient to assume a mean

value for^d'. Moreover, since p^ enters into equations (27)

only in terms which are multiplied by a; and therefore have

but smg,ll influence on the value of the work done, we mayalso substitute for p,, the most probable value for p„'.

The pressure p^', which exists in the waste space, depends,as already mentioned, on whether the shutting off from the

condenser takes place before or after the end of the stroke,

and may therefore have very different values. But this

pressure, and the quantities depending on it, occur in equa-

tions (27) only in terms containing the small factors /j, and

fj,„,so that an approximate value is sufficient for our purpose.

In cases where there are no special circumstances which

cause p^' to vary widely from p^', we may neglect this differ-

ence, as we did that between p„ and p/, and may thus assume

as a general value for all three quantities the most probable

mean value for the back-pressure within the cylinder. This

value we may call p^.

These simplifications change equations (27) into the

following



(28).

APPLICATIDir TO THE STEAM-ENGINE. 257

W =mj>, - m,p, + MG {T, - T,) + fi^, - f,C {T,-T,)]

m^. = «».?. + MO (T, - T,) + ^„p„ -,iG{T,- T,)

+ /ôWo {jPi -P,) + Ma- {p^-p^

§ 14. Substitution of the Volwme for the porresponding

Temp&rature in certain cases.

In the above equations it is assumed that, besides themasses M,m^, fju, and /*,,, of which the two first must be found

by direct observation, and the two latter can be approxi-

mately determined from the size of the waste space, we havealso given the four pressures p^, p^, p„ and p„, or, which is

the same thing, the four temperatures T^, T^,- 1\, and T^,

In practice however this condition is only partially fulfilled,

and we must therefore bring in other data to assist us.'

Of the four pressures here mentioned, two only, p^ and p^,may be taken as known : of these the first is given directly

by the gauge on the boiler, and the latter can be at least

approximately fixed by the gauge on the condenser. Thetwo others, p^ and p^, are not given directly ; but we knowthe dimensions of the cylinder and the point of cut-ofi", and

can thence deduce the volume of the steam at the moment

of cut-off and at the end of the stroke. We may then takethese volumes as our data in place of the pressures p^ and p^.

For this purpose we must throw the equations into a form,

which will enable us to use the above data for calculation.

Let us now, as before in the explanation of Pambour's

theory, denote by v' the whole space within the cylinder

which is open to the steam during one stroke, including the

waste space; by ev' the space opened to the steam up to the

moment of cut-off; and by e/ the waste space. Then in the

same way as before we have the following equations

mijj ttj -f- (i¥-t- /t) o- = ev',

c. 17



258 ON THE MECHANICAL THEOST OF HEAT.

The quantities fjb and a are both so small that their pro-

duct may be at once neglected ; whence we have

m^ u^ = ev' — Ma

m, u, = v' — Mcr

/*o=

ev.(29).

Further, we have by equation (13) of Chapter VI.

P = Tug,

writing the single letter g for -^, which Vyill occur very

frequently in what follows. We can therefore replace p^ and

Pjin equations (28) by u^ and u^. Then the masses m, and

TWj occur only in the products vnû^ and mû^, for which we

may substitute the values given in the two first of equa-

tions (29). Again, by the last of these equations we mayeliminate the mass fi^; and as regards the qther mass n,

though this may be somewhat larger than fi^, yet, since the

terms which contain (i are generally insignificant, we can

without serious error use the value found for /t„: in other

words we may drop for the purposes of calculation the as-

sumption made for the sake of generality, that the original

mass in the waste space is partly liquid and partly gaseous,

and consider the whole of it to be in the gaseous form.

The above substitutions may be effected in the more

general equations (27), as well as in the simplified equations(28). This substitution presents no difficulty, and we will

here confine ourselves to the latter set, in order to have the

equations in a form adapted for calculation. With these

changes they read as follows :

W' = mJ>, + MCC^,-T,)-{v'-M<y){T,g,-p^^p:)^

+ SV,p^-C(T,-T,)

{ev' - Ma) T^, = m,p, +MG {T, - T,)

{v'-Ma)g,= (ev'-Ma)g^+(M + '£)Clog^

(30).



APPLICATION TO THE STEAM-ENGINE.

§15. Work per Unit-Weight ofSteam.

259

To adapt the above equations, which give the work perstroke or per quantity of steam »w,, to determine the

workper unit-weight of steam, we must apply the same methodas before in transforming equation (17) into (18): viz. to

divide the three equations by m,, and then put

^ = ?, ^=F,andI^=F;m m,. TO.

then the equations become

M„

(p,-C{T,-T,)+er

f

(31).

(V-I<T)g, = {er-la)g,+ (l+'£) Clog^

§ 16. Treatment of the Equations.

The application of the above equations to calculate the

work done may be effected as follows. Assuming the pres-

sure of steam to be known, and also the speed at which the

engine works, we can thence determine the volume V of one

unit-weight of steam. By help of this value we first calcu^

late the temperature T^ from the second equation, then T^

from the third, and finally apply these to determine the work

done from the first equation.

Here however we are met by another difficulty. In order

tocalculate T^

andT,

fromthe

twolatter equations,

thesemust be solved for those temperatures. But they contaio

those temperatures not only explicitly but also implicitly,

since p and g are functions of temperature. If to eliminate

these quantities we use for p one of the ordinary empirical

expressions for the pressure as a function of tempera-

ture, and for g the differential coefficient of the same ex-

pression, then the equations become too complicated for

17—2





-APPLICATIOK TO THE STEAM-ENGINE. 261

bourhood of 100°, owing to his having employed for the

calculation of the constants logarithms to seven places of

decimals only. Moritz has therefore recalculated these con-

stants fromthe same observed values, but using ten places

of logarithms, and has published the values oip derived from

this improved formula, so far as they differ from Regnault's

values, which first takes place at 40°. These are the values

used by the author*.

When g has been calculated for the various temperatures,,

the product Tff can be calculated without further difficulty,

since T is given by the simple equation

The values of g and Tg thus found are given in a table

at the end of this Chapter. To complete this the corre-

sponding values oip are added, those from 0° to 40° and above

100° being Regnault's numbers, and those from 40° to 100°

being Moritz's. By the side of each of these three columns are

given the differences between each two successive numbers,

so that this table enables the values of these thrqe quantities

to be found for any given temperature, and conversely the

temperature corresponding to any given value of one of these

three quantities.

§ 18. Introduction of other Measures of Pressure and

Heat.

One other remark is to be made as to the mode of usingthis table. In equations (31) it is assumed that the pressure pand its differential coefficient g are expressed in kilograms

per square metre, whereas in the table the same unit of

pressure is employed as in Regnault's tables, viz. a milli-

metre of mercury. Hence, if in the formulae which follow

we are to consider p. and g as expressed in these latter iinits,

we must alter equations (31) by multiplying p and g by the

number 13"596 which expre3ses the specific weight of mercury;

inasmuch as this, by Chapter yi,, § 10, is the ratio between

the units. If we denote this number by A;, we must sub-

stitute hp and hg for p and g, whenever these occur in the

above equations. Similarly for the quantities G and p, which

express the specific heat and the heat of vaporization in,

* See note at end of chapter.



262 ON THE MECHANICAL THEOKY OF HEAT.

mechanical units, we will use c and r which, refer to the ordi-

nary unit of heat. For this purpose we must writers and-

Er for G and p.

Ifwe now divide the equations by h, so as to bring the con-stants E and k together as much as possible, these equations ,

Wtake the following form, from which to calculate -j-

thereby obtain W, the work done :

~ = ~[r, + h{T,-TJ\-{V-l<T){T,g,-p,+p,)

and

k M„

E,

{V-W)g,= {eV-la)9,+ (l + '-Pl^\og^

(32).

.(33).

EThe quantity y has the following value

^_ 423-55 _&-l3^ = ^^^^2^

§ 19. Determination of the Temperatures Tj, and T

The second of equations (32) may be written as follows

T^,= 0+a(t,-t^-b{p,-p^)(34),

where C, a and 6 are independent of T^, and have the values

_E

6=eV-la-

ev-y

(34a)._




Of the three terms on the right-haud side of (34) the first

is by far the most important ; and it is thus possible to

determine T^g^, and thereby also the temperature t^ by

successive approximations. To obtain the first approximationto Tg, -which, we may. call T'g', we may write t^ for t^, and

Pi for^j. Then we have

T'g'^G (35).

The temperature t' corresponding to this value of Tgmust be found from the table. Then to obtain the second

approximation to Tg we may write the value H thus found

for t^ in equation (34), and the corresponding value

pof the

pressure for p^. Thus, remembering equation (35), we have

for the second approximation

T"g"= Tg' + a{t,~ t') - h {p~p') (35a).

The temperature t" corresponding to this value of Tg

must be found as before from the table. If a second ap-

proximation be not sufficient, the process must be repeated.

Writing in equation (34) t" andp" in the place of i, andp,,

and remembering equations (35) and (35a), we have

T"'g"'=T'g" + a{t'-t")-b{p' -p") (356).

The new temperature t'" can again be found from the table.

In this way we may proceed to any degree of approxima-

tion ; but the third approximation only differs by about-^ ofa

degree, and the fourth by less than-j^Vir

°^ ^ degree, from the

true value of t^.

The treatment of the third of equations (32) is very

similar. If we divide by F— la, and for convenience of calcu-

lation replace the natural by common logarithms (for which

purpose we have to divide by the modulus M) the equation

takes the form

g=G+a'Log^.... (36),

where C and a have the following values independent of T^ :

„ eV-la-

a = T- Xk M

(V-W) J

.(3.6a).




In equation (36) the first term is again tlie mosi im-

portant, and we can therefore proceed hy successive approxi-

mations. First write T^ for 1\; we then have as the first

approximation for cr„

g'=C (37).

The corresponding temperature t' can be found from the

table, and from this we can easily obtain the absolute tem-

perature T'. Write T' for T^ in equation (36); then we have

Sr"=5r' + aLogp (37a).

This gives T". Similarly we can obtain the equation

g"'=g" + a\og^ (376).

-virhich again gives T"; and so on. Here again a very few

approximations will give a value which very closely agrees

with the actual value of 2^.

§ 20. Determination, of c and r.

,It now only remains to determine the quantities c and r,

before proceeding to the numerical application of equations

The quantity e, or the specific heat of the liquid, has

hitherto been treated as constant. This is not perfectly correct,-

since it increases slightly as the temperature increases. If

however we take as its general value the correct value for

the mean ofthe temperatures which occur in the investigation,

the divergencies may be neglected. This mean temperature

in the case of engines driven by steam may be taken at 100°,

which for an ordinary high-pressure condensing engine is

about a mean between the tempeirature of the boiler and that

of the condenser. Taking therefore the value of the specific

heat which Regnault gives for water' at 100°, we may put

c = l-013 (38).

To determine r we start from the equation which Regnault

gives for the whole quantity of heat required to heat a

unit-weight of water from 0° to the temperature t, and to

transform it at that temperature into steam. This equation is

A, = 606-5 + 0-305^



Jo

APPLICATION TO THE STEAM-ENGINE. 2&5

If here we substitute for \ the expression corresponding

to the foregoing definition, viz. I cdt + r, we haveJo

r = 606-5 + 0-305f- f cdfc

Jo

In the integral we must use for c the function of tem-

perature exactly detetmined by Eegnault*, if we are to

obtain for r the precise values which Regnault gives. For

the present purpose it is perhaps sufficient here also to use

for c the constant above-mentioned. This gives

rt

odt=l-01St;la

and the two terms involving t in the last equation can now

be combined into one, viz. — O'lOSt. We must at the same

time make some change in the constant term of the equa-

tion. This we will so determine, that the same value of r

which is probably the most accurate of all given by observa-

tion, shall also be correct as given by the formula. Nowat 100° Regnault found for \, as the mean of 38 observations,

the value 636 67. If from this we subtract the quantity of

heat required to heat the unit-weight of water from 0° to

100°, which according to Regnault is lOb'5 heat-units, there

remains to one place of decimals only

r^ = 536-2t.

Applying this value we obtain for r the formula

r = 607-0-708« (39).

This formula is already laid down in Chapter VII., § 3

and a short table is there given, which exhibits the close

accordance between the values of r as calculated by this

formula, and as given by Regnault in his tables.

§ 21. Special Form of Equation (32) for an Engineworking without expansion.

In order to distinguish the effects of the two different

kinds of expansive action, to which the two latter of equa-

* Relation dee expiriencei. Vol. I. p. 748.

f Begnault himself used in his tables the nnmher 536-5 instead of the

ahdve; this simply arises from the fact that he has used the round number

637 for X at 100", instead of 636-67 as given above.




tions (32) refer, it seems desirable first to consider an engine

in which only one of the two occurs. We will therefore

begin with an engine that works without expansion. In this

case we may substitute for e, or the ratio of the volumes

before and after expansion, the value 1 ; and may also put

Tj = T^. The equations (32) are thus greatly simplified.

The last becomes an identity, and is therefore useless. In

the second the right-hand side remains unaltered, whilst the

left-hand side becomes {V—la)T^^. Finally the first

equation takes the form

h Mj

If we here replace {V-la) T^g^ by the expression on the

right-hand side of the second equation, all the terms con-

. .

Etainmg ^ as factor cancel each other, as do two terms con-

taining Zo- as factor, and the terms which remain can be

collected together as two products. Then the two equations

are

W— = F (1 - e) (p, -jp„) - la- (p, -p,)

(V-l<r)T,ff, = f{r, + lc{T,-T,)]

+ ^^{f^''°

"^Z^'^ +P^-Po}+^(P-P^)_

(40).

The first of these equations is exactly the same as is

given by Pambour's theory, if in equation (18) we put e = 1,

and then by means of equation (12). (having first put e = l

v' .

and — = F in that equation) replace B by the volume V.

The only difierence therefore is in the second equation, whichreplaces the simple relation between volume and pressure

assumed by Pambour.




§ 22. Numerical Values qf the Constants.

The quantity e which occurs in these equations, and re-

presents the ratio of the waste space to the whole space left

open to the steam, may be taken as 0"05. The quantity ofwater which the steam carries over with it into the cylinder

is different in different engines. Pambour considers that it

averages 0"25 of the whole mass entering the cylinder in the

case of locomotives, but much less, 005, in the case of station-

ary engines. We will here take the latter value, which gives

1 : 0"95 as the ratio of the whole mass entering the cylinder

to that part of it which is in the form of steam. Further,

let the pressure in the boiler, or p^, be five atmospheres, to

which corresponds the teiiiperature 152'22°; and let it be

supposed that the engine has no condenser, or, which is the

same thing, a condenser at the pressure of one atmosphere.

The mean Isack-pressure in the cylinder is then greater than

one atmosphere. With locomotives this excess of back-

pressure may be considerable, for the special reasons already

mentioned ; with stationary engines it is insignificant. Pam-

bour, in his tables for stationary non-condensing ,engines, has

altogether neglected this excess; and since our present object

is to compare the new formula with his, we will follow his

example and put^^ = one atmosphere.

We have then in this case the following values to insert

in equations (40)

e = 005

'=m=^''' .(41).

p, = 3800

^0 = 760

We may also fix once for all the following values

k= 13-596,

<r = 0001.

There now remain only the quantities V and jp, undeter-

mined in equations (40), in addition to the quantity W, of

which we are seeking the value. Of these V is the volume

of steam in the cylinder per Unit-weight, and p^ is the final

pressure within the cylinder.




§ 23. The least possible Value of V, and the correspond-

ing amount of Work.

We must first enquire what is the least possible value

of V. This value corresponds to the case in which thepressure in the cylinder is the same as in the boiler, and we

have therefore only to substitute p^ for p^ in the last of

equations (40). Thus we obtain




.niuin density a volume greater than is allowed by the

mechanical theory of heat; this view finds its expression in

equation (116). If by means of this latter volume we deter-

mine the work from Pambour's equation upon the twoassumptions that 6 = and that e =^'0'05, the values obtained

are 16000 and 15200. These quantities, as follows immedi-

ately from the greater volume, are both greater than those

found above, but not in the same ratio, because the loss of

steam due to the waste space is smaller accojrding to our

equations than according to Pambour's theory.

§ 24. Calculation of the Workfor other Values ofY.

"With an engine of the kind here considered, the efiS-

ciency of which was investigated by Pambour, the speed

which the engine actually assumed bore a proportion in one

case of 1*275 : 1, and in another, with a lighter load, of1"70 : 1, to the minimum speed as calculated by his theory

for the same intensity of evaporation, and for the same

boiler-pressure. To these speeds correspond in our case the

volumes 0"495 and 0"660 respectively. We will now, as an

example of the mode of determining the work done, choose a

speed which lies between these two, say in round numbers

F=0-6.

Our first business is now to find the value of T,^ corre-

sponding to this value of V. For this purpose we use equa-tion (34), which takes the following special form

T,g^ = 26-577 + 56'42 («, -^ - 0-0483 (p, -p,) ... (43).

If by means of this equation we obtain successive approxi-

mations to t^, as described in section (19), the series of values

is as follows

tf =13301,if' =134-43,

r =134-32,

Further approximations would differ only in higher places of

decimals, and if we confine ourselves to two places we may



270 ON THE MECHANICAL THEORY OB* HEAT.

takB the last number as the true value of t^t The corre-

sponding pressure is

^, = 2308-30.

If we substitute in the first of equations (40) these values

of V and p^, together with the approximate values of con-

stants as fixed in § 22, we obtain

pr= 11960.

By Pambour's equation (18) we have for the same volume O'C

W= 12520.

In order to exhibit more clearly the dependence of the

work upon the volume, and at the same time the divergence

between Pambour's theory and the author's on this point,

calculations similar to that for volume 0"6 have been made

for a Series of other volumes increasing by equal intervals.

The results are given in the following table. The first hori-

zontal row of numbers, which is separated from the rest,

contains the values for an engine without waste space. The

further arrangement of the table is easily understood.




is going on during the admission, the amount of steam re-

mains al-vâys the same as it was at the commencement;

while, according to the author's theory, a part of the priming

water is afterwards evaporated, and this quantity is greaterthe greater the expansion.

§ 25. Wcyrk done for a given Value of Y hy an Engine

working with Expansion.

We will now treat in the same way an engine working

with expansion, and will choose as our example a condensing

engine. To fix the degree of expansion, we will assume that

the steam is cut off at one third of the stroke. We then

have, to determine e, the equation

6-6=^(1-6);

whence, giving to e the value O'OS, we have

e =?:J

= 0-3666.

o

Let the boiler-pressure be as before five atmospheres. The

condenser-pressure can with good arrangements be kept

below -j^jj atmosphere. But since it is not always so small,

and in addition the back-pressure in the cylinder somewhat

exceeds the pressure in the condenser, we will assume the

mean back-pressure p^ to be in round numbers \ atmosphere,

or 152 millimetres. Tothis corresponds the temperature

ta— 60-46°. If lastly we give to I the same value as before,

the constants to be used in our example will he as follows

6 = 0-36667

e = 0-05

i= 1-053

>,

= 3800

^^0=152

(44).

To be able to calculate the work done, we only now need

id know the value of V. To assist us in choosing this,

we must first know the least possible value of V. This is

found exactly as in the case of the engine without expansion,

by putting p^^ for p^ in the second of equations (32), and




altering the otter magnitudes, whicli' are ' in combination

withPj,

in the same way. We thus obtain in the present

case the value 1-010. Starting from this, we will first

suppose that theactual

speedof the engine exceeds the least

possible in the ratio 3 : 2. We may then put in round

numbers V= 15, and we will calculate the work done at this

speed. .

We first determine the two temperatures t^ and t^, by

substituting this value of V in the two last of equations (32).

Of these t^ has already been approximately found for the

non-condensing engine; and since the present case only

differs from the former inasmuch as the quantity e, whichwas there taken equal to 1, has here a different value, we

need not go through the process again, but will merely give

the final result, which is

t, = 137-43»,

To determine tg we have equation (36), which takes in

this case the form

^3 = 26-604 + 51-515 Log ^ (45).

From this we obtain the following successive approxima-

tions :

t' =99-24° T

t" =101-93

t'" =101-74t"" = 101-76 J

The last of these, from which further approximations would

only diverge in higher places of decimals, we will consider

to be the correct value of <j, and will apply it, together with

the known values of t^ and i„, to the first of equations (32),

Thus we obtain

W= 31080.

If, taking the same value for V, we calculate the work

done by Pambour's equation (18), remembering to determine

B and b not from equation (115) as with the non-condensing

engine, but from equation (lla) -which refers to condensing

engines, we find

TF= 32640.



APPUCATION TO THE STEAM-ENGINE. 273

§ 26. Swmmary of Various Cases relfiting to the Work-',

ing of the Engine.

The work done for the volumes 1*2, 1'8 and 2'1 has

been calculated in the same way as described above for the

volume 1'5. Further, in order to make clear by an example

the influence, which the different imperfections of an engine

have upon the amount of the work done, the following cases

are subjoined.

(1) The case of an engine which has no waste space,

and in which the cylinder pressure is equal to the boiler

pressure, and the expansion carried so far that the pressure

has fallen from its original value p^ to p„. If we only assume

further that p^ is exactly equal to the condenser pressure,

this case is that to which equation (9) relates, and which gives

the greatest possible amount of work for a given quantity

of heat, when the temperatures at which the heat is taken

in and given off are also given.

(2) The case of an engine, where there is again no wastespace, and the cyhnder and boiler-pressures are equal, but

where the expansion is not as before complete, but is only in

the proportion of e : 1. This is the case to which equation (6)

refers ; except that there, in order to determine the magnitude

of the expansion, the change of temperature in the steam

caused thereby was assumed to be known, whereas here the

expansion is fixed by the volume, and the change of tem-

perature must be calculated therefrom.

(3) The case of an engine having waste space and in-

complete expansion, in which the only one of the above

favourable conditions which remains is the fact that the

steam in the cylinder has during admission the same pres-

sure as in the boiler, so that the volume has its least possible

value. To this are appended the cases already mentioned,

in which this last favourable condition is also absent, and

the volume has some other given value instead of the least

possible.

For purposes of comparison Pambour's theory has also

been applied to all these cases except the first, for which

equations (11a) and (llS) do not hold, inasmuch as even

the equation which he has determined for low pressures can

C. 18



274 ON THE MECHANICAL THEOBT OF HEAT.

only be applied to ^ or at the lowest ^- of an atmosphere,

whereas here the pressure has to descend to ^ of an atmo-

sphere. For this first case the numerical values given by our

equations are as follows

Volume before expansion = 0-3637,

after. „ =6-345,

Work done = 50460.

The result for all the other cases are contained in the

foUpwing table ; in which a line is drawn between the values

for an engine without waste space and the remaining values.

For the volume are given only the values after expansion,

because those before expansion, can be at once derived from,

these by diminishing them in the ratio 1 : e or 1 : 0-36667.

r




§ 28. Friction.

Something must be said in conclusion on tlie subject of

friction. This will be confined to a justification of the course

adopted in leaving friction altogether out of account in the

equations hitherto developed. For this purpose we will shewthat instead of introducing, friction at once into -the first

general expressions for the work, as Pambour has done, the

same principles will allow us to take it into account sub-

sequently, according to the method already adopted by other

authors.

The forces which the engine has to overcome during its

stroke may be distinguished as follows : (1) The resistance

which it meets with from without, and in overcoming which

consists its useful work. Pambour has named this the load

(charge) of the engine. (2) The resistances which originate

within the machine itself, so that the work expended in

overcoming them produces no useful effect. These latter

resistances we combine under the name of Friction, although

in strictness they comprise other forces than those of friction,

especially the resistances of the various pumps driven by the

engine, with the exception of the feed-pump, the effect of

which has already been considered.

Both these kinds of resistances Pambour takes into ac-

count as forces which oppose the motion of the piston ; and

in order to be able to combine them readily with the pres-

sures of the steam on both sides of the piston, he made his

method of denotation agree with that used in the case of the

steam pressures, so that the symbol denotes not the whole

force, but the force referred to one unit of surface of the

piston. On this understanding let the load be represented

byJi.

A yet further distinction must be made with regard to

the friction. This has notthe same constant value for

anyone engine, but increases with the load. Pambour divides

it therefore into two parts, that which exists when the engine

runs unloaded and that which is brought into action by

the load. The latter he takes to be proportional to the load

itself. He accordingly expresses the friction per unit of

surface by f+ BR, where / and S are quantities which

depend on the construction and dimensions of the engine,

18--2




but for any one engine may according to Pambour be con-

sidered as constant.

We may now refer the wort done by the engine to these

resisting forces, instead, as hitherto, to the moving forceof the steam. The negative work done by the former must

equal thi§ positive work done by the latter, because otherwise

an acceleration or retardation of the speed woiild take place,

and this is against our original assumption that the speed is

uniform. Whilst one unit-weight of steam enters the cylin-

der, the piston passes through the space (1 — e) V, and we

thus obtain the following expression for the work W: '

F=(l-e) V[{l + B)B+f].

The useful part of this work, which for distinction we may

call {W), is expressed as follows :

(W) = {l-e)rxB.

If we eliminate R from this equation by means of the one

above, we obtain

^^^_wâ^ mBy help of this equation, since V is assumed to be known,

we can deduce the useful work (W) from the whole work

W, as soon as/ and S are given us. The manner in which

these are determined by Pambour will not be entered upon

here, since that determination rests upon insecure grounds,and friction is not the special subject of this chapter.

§ 29. General Investigation of the Action of Thermo-

dynamic Engines and of its Relation to a Cyclical Process.

We have now so far completed our treatment of the

steam-engine, that we have followed out all the processes

which take place, determined the several positive or nega-

tive quantities of work performed therein, and combined

these into one algebraic expression. We will proceed to

consider thermo-dynamic engines from a more general point

of view. The expression that an engine is driven by heat

must not of course be understood of the direct action of the

heat, but only that some substance within the engine sets

the working parts in motion through the changes which it




undergoes by means of heat. This substance we will call the

transmitter of the heat's action.

If a continuously working engine is at uniform speed,

all the changeswhich occur must be periodic, so that thesame condition, in which the engine and aU its parts may be

found at any given time, will recur at regular intervals.

Therefore the substance transmitting the heat's action must

also at such recurring moments exist within the engine in the

same quantity and under the same conditions. This may be

accomplished in two different ways. (1) The same original

quantity of this substance may always remain within the

engine; in which case'the changes which it undergoes duringthe motion must be such, that at the end of each period it

returns to its original condition, and the same cycle of

changes then begins anew. (2) The engine may get rid at

the end of each period of the substance which during that

period has served to produce its action, and may then take

in the same quantity of the same substance from without.

This latter method is the more usual in practice. It ex-

ists, e.g. in the hot-air engine, as hitherto constructed,, since

after every stroke the air which has driven the piston in the

working cylinder passes out into the atmosphere, and is

replaced by an equal quantity of air drawn from the atmo-

sphere by the feeding cylinder. The same is the case with

the non-condensing steam-engine, in which the steam passes

from the cylinder into the atmosphere, and is replaced by

fresh water, pumped from a reservoir into the boiler. It

also applies, partially at least, to the ordinary condensing

steam-engine. In this the condensed water is indeed in part

pumped back into the boiler, but not as a whole, because

it mixes with the cold water, a part of which also enters

the boiler. The part of the condensed water which is not

used again, together with the remainder of the cold water,

must go to waste.

The first method has hitherto been applied in a few

engines only, amongst others in engines driven by two dif-

ferent vapours, e.g. steam and ether*. In these the steam

is condensed by contact with metal pipes filled inside with

liquid ether, and is then wholly pumped back into the

* Anmles; des Mines, 1853, pp. 203, 281.




boiler. Similarly the ether vapour is condensed in metal

pipes surrounded by cold water, and is then pumped back

into the vessel used for evaporating the ether. To keep up

a uniform working, it is therefore necessary merely to addso much water and ether as is lost by leakage.

In an engine of this kind, in which the same mass of

vapour is continually used anew, the different changes which

this mass undergoes during each period must, as shewn above,

form a complete circuit, or in the terms of this treatise a

cyclical process. On the contrary the engines in which a

periodical admission and emission of vapour takes place are

not necessarily subject to this condition. Nevertheless they

may fulfil it, provided that they emit the vapour in the same

condition in which they received it. This is the case with a

condensing steam-engine, in which the steam passes away

from the condenser as water and at the same temperature at

which it passed from the condenser into the boiler. The

waste water, which enters the condenser cold and leaves it hot,

need not be taken into account, because it does not appertain

to the substance transmitting the heat's action, but merely

serves as a negative source of heat.

With other engines the conditions at the emission and

admission are different. For example, hot-air engines, even

if fitted with a regenerator, send the air back to the atmo-

sphere at a higher temperature than that at which it enters

and non-condensing engines take in water and emit steam.

In these cases no complete cyclical process takes place ;

butit is always possible to conceive that the actual engine has

another attached to it, which receives the vapour from the

first, restores it in some way to its original condition, and

then lets it escape. The two can then be considered as one

engine satisfying the above condition. In many cases the

process can be completed in this way, without making the

investigation too complicated ; e.g. we may suppose that a

non-condensing steam-engine, assuming that the feed water

is at 100°, is replaced by an engine having a condenser, the

temperature of which is 100°.

We may thus apply to all thermo-dynamic engines the

principles which hold for a cyclical process, if we only sup-

pose those engines which do not of themselves fulfil the con-

dition, to be completed in the manner described : and we can





280 ON TBE MECHANICAL THEORY OF HEAT.

The second equation thus becomes

/,

whence

Again, since Q= Q^ + Q„, the first equation gives

or, substituting for Q„ its value just found,

W=^Q,-T,j^'^-T,N ..(48)..

If we make the special assumption that the whole process

is reversible, then N= 0, and the above equation becomes

W=Q,-T,j^'^^9 ^''^-

This equation diflfers from the former only in the term

— T^N. Since iVis always positive, this term must be nega-

tive ; whence we see, as is easily proved directly, that under

the conditions fixed above with respect to the imparting of

the heat, the greatest possible amount of work is obtained

when the whole cyclical process is reversible ; and that every

circumstance, which causes the changes in a cyclical process

not to be reversible, diminishes the amount of work done.

Equation (48) therefore leads to the required value of

the work done in a way which is the exact opposite to the

ordinary ; for we do not, as in other cases, determine the

several quantities of work done during the several changes,

and then add them together, but we start from the maxi-

mum of work, and then subtract from this the losses of work

due to the various imperfections of the process. This me-

thod may therefore be called the method of subtraction. If

we limit the conditions under which the heat is imparted, bysupposing that the whole quantity Qj is imparted at a given

temperature T,, then the other part of the integral also




becomes, immediately integrable, and gives -—. Equation

(47) then takes the following form for the maximum work :

T -T

§ 31. Application of the above Equations to the Limiting

Case in which the Cyclical Process in a SteamrMngine is

Among the cases considered above with regard to the

action of the steam-engine, there is one which cannot indeed

be attained in practice, but which it is desirable to approach

as nearly as possible, viz. the case in which there is no waste

space, in which the cylinder-pressure is the same as in the

boiler or condenser respectively, and in which the expansion

extends so far, that the steam is thereby cooled from the

temperature of the boiler to that of the condenser. In this

case the cyclical process is reversible in all its parts. Let ussuppose that evaporation takes place in the condenser at the

temperature T^ ; that a mass M, of which the part m^ is in the

gaseous, and the part M—m^ in the liquid form, passes into

the cylinder and drives the piston upwards ; that by the fall

of the piston the steam is first compressed until its tempera-

ture rises to T^ , and then at that temperature forced into the

boiler ; and that lastly the mass M is again brought out of

the boiler in the liquid form into the condenser, by means of

the small pump, and there cools to the initial temperature T„.

Here the steam passes through the same series of conditions

as before, but in the reverse order. The various quantities

of heat are imparted or withdrawn in the opposite sense, but

to the same amount and at the same temperature of the

mass ; and all the quantities of work have the opposite signs

and the samenumerical values. Hence it follows that in

this case there is no uncompensated transformation in the

process. We may therefore put iV=0 in equation (48),

thereby obtaining an equation similar to (49), but in which,

for the sake of distinction, we will write W instead of W,

viz.

^'=«.-<'f-




Here Q, is in our case the heat imparted within the boiler to

the mass M, whereby M is heated as liquid from T^ to Z\,

and then the part m^ transformed into steam. Hence

Q, = m,p, + MC{T,-T,) (51).

To determine | -^ we must consider separately the

two quantities contained in Q^, viz. MG (T^ — T^ and»»,Pj.

To integrate for the first of these, let us write the element

dQ in the form MCdT: then this part of the integral

becomes

MO ^= MClog^.

Whilst the latter quantity of heat is being imparted, the

temperature remains constant at Tj, and the integral in this

case is simply

Substituting these values, the expression for W becomes

This is the exact expression given in equation (9), and found

in sections 4 and 5 by the successive determination of the

several quantities of work done during the cyclical process.

§ 32, Anothei" Form of the Last Expression.

It was observed in the last section with reference to

equation (51) that the two quantities of heat which make up

Q^, viz. m^p, and MG (T, — r„), must be treated differently

in the calculation of the work done ; inasmuch as the one is

imparted to the substance which transmits the heat's action

at a fixed temperature T^, and the other at temperatures

rising continuously from T^ to Tj. Similarly these two




quantities of heat occur also in different forms in the expres-

sion for the work, as is more clearly shewn if we write ourlast equation as follows

W = m,p,^^'' + MG{T,-T,) (l + ^^^^bg Jo)...(52).

Here m^pj^ is multiplied by the factor

which occurs in equation (50), while MO (T^ — T^ is multi-

plied by the factor

T Tl + TrfvTlog-^.

In order to compare these factors better with each other,

we will throw the latter into another form. If for brevity

we write

T -T«=-V-" (^3)'

and we thus obtain

- l — zfz z^ z^ , \

=l-^(l + 2+-3 + ^*°-)

Z , 1^ , z^+ 75—iT + j^—7+ etc.1x22x33x4

Equation (52) or (9) thus becomes

F' = m^,x.-Filfa(T,-r„)x.(i+2^H-3?^+etc.)

(54).

The value of the infinite series in the bracket, which

makes the difference between the factor oiMC (T^ — Tq) and

the factor of m^ p,, varies, as is easily shewn, between ^ and 1,

whilst z increases from to 1.




§ S3. Influence of the Temperature of the Source of

Heat.

Since, under the assumptions we have made, the cyclical

process passed through periodically by the engine during its

motion is reversible in all its parts, as shewn in § 31, and

since a reversible process gives the maximum work attain-

able, we are able to lay down the following principle :

If the temperatures at which the substance transmitting

the heat's action takes in the heat from the source, or gives

it out externally, are taken as given beforehand, then the

steam-engine, under the assumptions made in the develop-ment of equations (9) or (52), is a perfect engine, inas-

much as with a given quantity of heat imparted to it, it

performs the greatest quantity of work, which according to

the. Mechanical Theory of Heat it is possible to perform at

those temperatures.

It is otherwise, if we consider these temperatures not as

given beforehand, but as a variable element which must be

taken into account in our estimate of the engine.

The fact that the water, whilst being heated and evapo-

rated, has a much lower temperature than the fire, and there-

fore the heat imparted to it passes from a higher temperature

to a lower, produces an uncompensated transformation which

is not included in JV of equation (47), but which greatly

diminishes the useful effect of the heat We see by equa-

tion(54)

that the work obtained by the steam-engine from

the quantity of heat m,p, -1- MG (T, — T^) = Q,, is somewhat

T —Tless than Q^—^—-"

. But if we assume that the substance

i

transmitting the heat's action could have, throughout the time

when it is taking in heat, the same temperature as those parts

of the fire which furnish that heat; and if we denote the

mean value of this temperature by T, and the temperature at

which the heat is given off, as before, by T^; then under these

circumstances the work which it is possible to obtain from

the quantity of heat Q, will by equation (50) be expressed

T — T^y Qi

—Tpr-^ • To compare the values of this expression in

various cases, let f„, the condenser temperature, be fixed at

50°, and let us take for the boiler temperatures 110°, 150°, and




180°; the two first of wliich about correspond to those of the

low-pressure and ordinary high-pressure engines, while the

last may be considered the limit of temperature, which has

as yet been practically attempted for the steam-engine. InT —Tthese cases —^—' has the following values (adding in each

case the number 273 to give absolute temperatures)

When T, = 110° -»^^» = 0-157,

....150° ! 0-236,

180" 0-287.

On the other hand if for example we assume t' at 1000°,

, T' — T .

the corresponding value of—™ - is 0-746.

Hence it is easy to see, as already observed by Carnot

and subsequent authors, that in order to get the greatest

advantage from engines driven by heat, the most important

point is to increase the temperature interval T — JJ,

For example the hot-air engine can only be expected to

shew a decided advantage over the steam-engine, when

the possibility is established of its being worked at de-

cidedly higher temperatures, such as are forbidden to the

steam-engine for fear of explosion. The same advantage

may however be obtained by the use of superheated steam,

since, as soon as the steam is separated from the water, it

may be heated with no more risk than a permanent gas.

Engines which use the steam in the superheated condition

ought to unite many advantages of the steam-engine with

those of the air-engine, and may therefore be expected to

achieve a practical success much sooner than the latter.

In the class of engines mentioned above, in which besides

the water another more volatile substance is used, the inter-

val T^ — Tq is widened, inasmuch as T^ is lowered. It has

been already suggested that the interval might be widened

at the higher limit in the same manner, by adding yet a

third liquid less volatile than water. The fire would then

directly evaporate the first of the three only ; this in con-

densing would eva,porate the second, and this in like manner

the third. As far as principle goes this combination would




undoubtedly be advantageous ; the practical difficulties, whicli

might hinder its being carried out, cannot be conjectured

beforehand.

§ 34; Example of the application of the Method of Sub-

traction.

The ordinary steam-engine, besides the imperfection

described above, which is inherent in its nature, possesses

many other imperfections, which are rather due to its con-

struction in practice : some of these have already been taken

into account in our investigations to determine the work done.

We will now in conclusion exhibit the method in which, withengines possessing such imperfections, the work done maybe determined by the method of subtraction given in § 30.

In order that we may not extend this investigation, which

is merely an example of the mode of applying this method, to

too great a length, we will only take into account two of

these imperfections, viz. the existence of the waste space and

the difference of the pressures in the cylinder and boiler,

during the admission of steam. On the other hand we will

suppose the expansion to be complete, so that the final tem-

perature Tj equals the condenser temperature T^; and we

will also take T„' and y„" as equal to ?"„.

The process is founded on equation (48), which, calling

the work W, is as follows

Here Q^ — T^\ -^ represents the maximum work, cor-

responding to the case when the cyclical process is reversible;

and T^ represents the loss of work due to the imperfections

which arise from the process not being reversible. For this

maximum work we have in the case of the steam-engine the

expression already given at theend of § 31,

viz.

It only remains to determine JV, the uncompensated

transformation which occurs in the cyclical process. This

transformation arises when the steam is passing into the

waste space and into the cylinder, and the data for its deter"



APPLICATION TO THK STEAM-ENGINE. 287

mination are already given in § 10. There, by assuming

that the mass of steam admitted was at once forced back

again into the boiler, and also that everything was brought

back by a reversible path to its initial condition, we arrivedat a special cyclical process, for which we determined all the

quantities of heat imparted to the variable mass, and to

which we can now apply the equation

^=-jfThese quantities of heat, some positive some negative, are as

follows

7n,p„ -mj>„f,,p„,

MG{T,- T,) and -/*C7(2;- T,).

The three first of these are imparted at the constant

temperatures T^, T^ and T^ respectively : and the correspond-

ing parts of the integral are ^\ -"^ and ^. The

two last are imparted at temperatures which vary contmu-

ously between T^ and T^, and between T^ and T^ respectively;

and the corresponding parts of the integral are

T TMC log-™' , and - fiC log -^^

If we substitute the sum of these quantities for the integral,

the last equation becomes-

jVr=_^. + ?M»_if01og|-^» + /^Clog|...(55).

Multiplying this expression for JV by Tj, and subtracting

the product from the expression given above for the maxi-

mum work, we obtain finally for W' the equation

W = mj>,-^ m^, +MC (T,- T,) - (M+fj) GT„ log| + ^L,p

(56).

To compare this expression for W with that given in the

first of equations (28), we have only to substitute in the

latter the value of m^pg given in the last of those equations,

and then put T, = r„. The expression thus obtained agrees

exactly with that in equation (56).




In the same way we may also deduct the loss of work

due to incomplete expansion. For this purpose we must cal-

culate the uncompensated transformation which arises during

the passagejof steamfrom the cylinder to the condenser, and

then include this in the expression for W. By this calcula-

tion, which we shaiU not here perform at length, we arrive

at the same expression for the work as is given in equation

(28).

NOTE ON THE VALUES OF ^.at

Since the differential coefficient — occurs frequently in researches

upon steam, it is important to know how far the convenient method of

determination employed by the author is allowable ; and for this

purpose a few values are here placed side by side for comparison.

The, formula employed by Eegnault to calculate the pressiures of

steam in his tables, for temperatures above 100°, is the following

Log^=a-ba' - c^,where Log denotes the common system of logarithms, m denotes the

temperature from -20" to the value under consideration, so that

a;=t + 20, and the five constants have the following values

a =6-2640348,

Log 6=0-1397743,

Log 0=0-6924351,

Log 0=9-994049292 - 10,

Log yS= 9-998343862 - 10.

From this formula we obtain the following equation for -^

where .a and /3 have the same values as before, and A and B have the

following values

Log 4=8-5197602 -10,

Log5= 8-6028403 - 10.

Suppose tbat from this equation we calculate the value of ^ for thedt

temperature 147°, then we obtain

m =\dtjw

90-115.



APPLICATION TO THE STEAM-ENGINE. .289

Next for the approximate mettod of determination, we obtain.from

Begnault's tables the pressures

îffi=3392'98,

î6=3214'74,

whence £yiZ£L«= l!2^ =90.13.2 2

This approximate valiie agrees so closely with the exact value deduced

above, that it may be safely used for it in all investigations upon the

steam-engine.

With regard to temperatures between 0" and ,100°, the formula used

by Begnault to calculate the pressures of steam between these' limits is

the following

log^=a + 6a'-c|3'.

Here the constants, by Morjtz's improved tables, have the following

values

a=4-7393707,

Log 6=8-13199071I2 -;10,

Logc=0-6X174Q7675,

Log 0=^0-006864937152,

Logi3=9-996726536856 - 10.

From this formula we can again obtain for -^ an equation of the form

where the values of a and ^ are the same as given above, and those of

A and B are as follows

Log.4= 6-6930586-10,

Log5= 8-8513123 -10.

If we calculate from this equation the value of ~ corresponding to a

temperature of 70", we obtain

=10-1112.

By the approximate method of determination we obtain

p„ -io^_ 243-380- 223-154_^^.

2 2

This result again agrees sufficiently closely with that obtained from

the more exact equation.

c. 19




Table giving the values for steam of the pressure p, its

differential coefficient -^ = g, and the product Tg, for

different temperatures: all expressed in millimetres of

mercury.

t in

degrees

Oiinti-

grade.




t in




t iw




tin



294' ON THE MECHANICAL THEORY OF HEAT.

t lU

centi-

grade.

159



CHAPTER XII.

ox THE CONCENTRATION OF KAYS OF LIGHT AND HEAT

AND ON THE LIMITS OF ITS ACTION.

§ 1. Object .of the Investigation.

The principle assumed by the author as the ground of

the second main principle, viz. that heat cannot of itself, or

without compensation, pass from a colder to a hotter body,

corresponds to everyday experience in certain very simple

cases of the exchange of heat. To this class belongs the

conduction of heat, which always takes place in such a way

that heat passes from hotter bodies or parts of bodies to"

colder bodies or parts of bodies. Again as regards the ordi-

nary radiation of heat, it is of course well known that not only

do hot bodies radiate to cold, but also cold bodies conversely

to hot ; nevertheless the general result of this simultaneous

double exchange of heat always consists, as is established by

experience, in an increase of the heat in the colder body

at the expense of the hotter.

Special circumstances may however occur during radia-

tion, which cause the rays, instead of continuing in the same

straight line, to change their direction ;

and this change of di-

rection may be such, that all the rays from a complete pencil

of finite section meet together in one point, and there combine

their action. This can he accomplished, as is well known, by

the use of a burning-mirror or burning-glass ; and several

mirrors or glasses may even be so an-anged, that several

pencils of rays from different sources of heat meet together iii

one point.




For cases of this kind there is no experimental proof that

it is impossible for a higher temperature to exist at the point

of concentration than is possessed by the bodies from which

the rays emanate. Rankine* accordingly, on aspecial occasion,

of which we shall speak in another place, has drawn a parti-

cular conclusion, which rests entirely on the assumption that

rays of heat can be concentrated by reflection in such a way,

that at the focus thus produced a body may be raised to a

higher temperature than is possessed by the bodies which

emit the rays. If this assumption be correct, the principle

enunciated above must be false, and the proof, deduced by

means of that principle, of the second fundamental principle

of the Mechanical Theory of Heat would thus be overthrown.

As the author was anxious to secure the principle against

any doubt of this kind, and as the concentration of rays of

heat, with which is immediately connected that of rays of

light, is a subject which, apart from this special question, is

of much interest from many points of view, he has attempted

a closer mathematical investigation of the laws which govern

the concentration of rays, and of the influence which this con-

centration can have on the exchange of heat between bodies.

The results are contained in the following sections.

I. Reasons why the ordinary method of deter-

mining THE mutual radiation OF TWO SURFACES

does NOT EXTEND TO THE PRESENT CASE.

§ 2. Limitation of the treatment to perfectly black bodies,

and to homaffeneous and unpolarized rays of heat.

When two bodies are placed in a medium permeable to

heat rays, they communicate heat to each other by radiation.

Of the rays which fall on one of these bodies, part is in gene-

ral absorbed, part reflected, part transmitted; and it is knownthat the power of absorption stands in a simple relation to

the power of emission. As it is not here our object to inves-

tigate the differences between these relations and the laws

to which they conform, we wiU take one simple case, viz.

that in which the bodies are such that they completely ab-

sorb all the rays which fall upon them, either actually on the

• On the Ee-concentration of the Meehanioal Energy of the Universe,P/iiJ. Mag., Series iv., Vol. it. p. 358.



CONCENTEATION OF BATS OF LIGHT AND HEAT. 297

surface, or in a layer so thin that its thickness may be neg-

lected. Such bodies have been named by Kirchhoff, in his

well-known and excellent paper on the relation between

emission and absorption, "perfectly black bodies*."Bodies of this kind have also the maximum power of

emission. It was formerly assumed to be beyond question

that their intensity of emission depended only on their tem-

perature; so that all perfectly black bodies, at the sametemperature and with the same extent of surface, would

radiate exactly the same quantity of heat. But as the rays

emitted by the body are not homogeneous, but differ accord-

ing to the scale of colours, the question of emission must be

studied with special reference to this scale ; and Kirchhoff has

extended the foregoing principle, by laying down that per-

fectly black bodies at equal temperatures send out not only

the same total quantity of heat, but also the same quantity

of each class of ray. As the distinctions between the rays

according to colour have no place in our investigation, we

will assume throughout that we have to do with only oneknown class of ray, or, to speak more accurately, with rays

whose wave-length only vaaries within indefinitely small

limits. Whatever is true of this class of rays must similarly

be true of any other class; and thus the results obtained

from homogeneous heat may easily be extended to heat

which contains a mixture of different classes of rays.

With the same object of avoiding unnecessary complica-

tions, we will abstain from discussing polarization, and assume

that we have only to do with unpolarized rays. The mode

of taking polarization into account in such cases has already

been explained by Helmholtz and Kirchhoff.

§ 3. Kirchhoff's formula for the mutwil Radiation of

two Elements of Swface.

Let Sj and s, be the surfaces of two perfectly black

bodies of the same given temperature ; and let us consider

two elements of these surfaces ds^ and ds^, in order to

determine and to compare the quantities of heat which they

mutually send to each other by radiation. If the medium,

which surrounds the bodies and fills the intervening space,

• Pogg. Ann., Vol. oix. p. 275.




is homogeneous, so that the rays simply passin straight lines

from one surface to the other, it is easy to see that the quan-

tity of heat which ds^ sends to ds^ must be the same as that

which ds^ sends to ds^ ; if on the contrary this medium is

not homogeneous, but there are variations in it which cause

the rays to be broken up and reflected, the process is less

simple, and a closer investigation is needed to prove whether

the same perfect reciprocity holds in this case also. This

investigation has been performed in a very elegant manner

by Kirchhoff; and his results will be briefly stated here, so

far as they relate to the ease in which the rays on their

way from one •element to the other suffer no diminution of

strength ; in other words in which the breakings up and re-

flections of the ray take place without loss of power, and

there is no absorption during its passage. A few variations

will alone be made in the denotation and in the system of co-

ordinates, to produce a better accordance with what follows.

If two points are given, only one of the infinitely large

number of rays sent out by one point can in general attain

the other*; or if the rays are so broken up and reflected,

that several of them mêt in the other point, yet they form in

general only a limited number of separate rays, each of which

can be treated by itself. The path of such a ray from one

point to the other is determined by the condition that the

time expended in traversing this path is a minimum, com-

pared with the times which would be expended in traversing

all other neighbouring paths between the same points. This

minimum time is determined, if where, several separate rays

meet we investigate only one at a time, by the position of

the two points between which it passes ; and we will denote

it, as Kirchhoff has done, by T.

* The form of expression tliat a point sends out an infinitely large num-ber of rays is perhaps, in the strict mathematical sense, inaccurate, since heat

and light can only be sent forth from a surface, and not from a mathematical

point. It would be more accurate to refer the sending out of the heat or light

not to the point itself, hut to the element of area at the point. As, however, the

conception of a ray is itself only a mathematical abstraction, we may, without

fear of misconception, retain the statement tiat an infinitely large number of

rays proceed from each point of the surface. K it were our objeet to determine

quantitatively the heat or light radiated by a surface, it is evident that the

size of the surface must be taken into account, and that its elements must he

considered, not as points, but as indefinitely small surfaces ; the area of

which must appear as a factor in the formula expressing the quantity of heat

or light radiated from ah element of surface.



CONCENTRATION OF rAtS OF LIGHT AND HEAT. 299-

Returning now to the elements of surface ds^ and dSj, wewill suppose a plane tangential to the surface to be drawn'

through one point of each element; and we will treat ds^ and

ds^ as elements of these planes. In each of these planes let us

take any system of rectangular co-ordinates, which we will

call aTj,2/1 in the one case, and os^, y^ in the other*. If we

now take a point on each plane, the time T^, expended by

the ray in passing from one point to the other, is determined,

as stated above, by the position of the two points; and this

time may therefore be treated as a function of the four co-

ordinates of thetwo

points.

On these assumptions Kirchhoffs expression for the

quantity of heat, which the element ds-^ sends to the element

rfsjj per unit of time, is as follows i";

•7r\i

d^'T d'T d'T d'T

dx^dx^ dy^dy^ dx^dy^ dy^dxj ' "'

where ir is the well-known ra.tio between the circumference,

and diameter of a circle, and e, is the intensity of emission

of the surface Sj, in the locality of the element ds^, so that e^ dsj^

represents the whole quantity of heat radiated by ds^ per unit,

of time.

Conversely to express the quantity of heat which ds^

sends to ds^, we need only substitute for e^ in the above,

expression the quantity e,, which is the intensity of emission

of the surface s,. Everything else remains unaltered, as

being symmetrical with regard to the two elements ; for the

time T, which a ray expends in traversing the path between

two points of the two elements, is the same in whichever

direction it is moving. If we now assume that the surfaces,

which are supposed to be at the same temperature,radiate eqiâl

quantities of heatin the same time, then 61 = e^ ; and there-

fore the quantity of heat sent by ds^ to ds^ is exactly the same

as that sent by ds^ to ds^.

* Kirohhoff has chosen two planes at right angles to the directions of the

rays in the neighbourhood of the two elements; he has taken axes of 00-'

ordinates in these planes, and has projected the elements of surface upon

them.

tPogg. Ann., Vol. oix. p. 286.





CONCENTRATION OF RATS OF LIGHT AND HEAT. 301

sidered as variable when difFerentiating according to «!,

whilst the second co-ordinate y^ of the same point, and the

co-ordinates x^, y^ of the other, are assumed to be constant,

and wh^re similarly in differentiating according to x^, this is

taken as the only variable—can thus represent no real quan-

tity of finite value. Therefore in this case we must find an

expression of somewhat dififerent form from Kirchhoff's ; and

for this purpose we must first consider some questions similar

to those considered by Kirchhoff in arriving at his expres-

sion.

II. Determination of corresponding points and coRr

RESPONDING ELEMENTS OF SURFACE IN THREE PLANES

CUT BY THE RAYS.

§ 5. Equations between the co-ordinates of the points in

which a ray cuts three given planes.

Let there be three given planes a, b, c (Fig. 25), and ineach of them let there be a system

of rectangular co-ordinates, which

we may call respectively oo^ya, x^y^,

and x^^. Let us take a point pîn plane a, and a point p^ in plane b,

and consider a ray as passing from

one of these to the other ; then to

determine its path we have the con-

dition that the time, which the ray

expends in traversing it, must be

less than it would expend in travers-

ing any other neighbouring path.

Call this minimum time T^y It is

a function of the co-ordinates of p^

and Pi,i.e. of the four quantities x^y^, x^y^. Similarly let

T^ be the time of the ray's passage between two points p^

and p„, in planes a and c ; and let T^^ be the time of its pas-

sage between two points p^ and p^, in planes b and c. T^ is a

function of the co-ordinates x^y^, x^^; and T^^ is a function

of the co-ordinates x^y^, x^y^.

Now as a ray, which passes between two of these planes,

will in general cut the third plane also, we have for each ray




three points of section, which are so related to each other

that any one of them is in general determined by the other

two. The equations which serve for this determination may

beeasily deduced from the above-mentioned condition. Let

us first suppose that, points p^ andp„ are given beforehand, and

that the point is still unknown in which the ray cuts the

intermediate plane h. This point, to distinguish it from

other points in the plane, we will call p\. Take any point

whatever p^ in this plane, and consider two rays, which we

may call auxiliary rays, passing the one from p^ to p^, and

the other from p^ to p„. In Eig. 25 these rays are shewn by

dotted lines, and the primary ray, which goes direct from p^to p^, by a full line*. If, as before, we call the times ex-

pended by these two rays T^^ and T^^, the value of the sum

of these times T„j + T^^ will depend on the position of jjj, and

therefore, since the points p^ and p^ are assumed to be given,

it may be considered as a function of the co-ordinates x^y^

of point Pj.Among all the values, which this sum may

assume if we give to point p^ various positions in the neigh-

bourhood of p 5, the minimum value must be that which is

obtained by making p^ coincide with p\, in which case the

auxiliary rays merely become parts of the direct ray. Wetherefore have the following two equations to determine the

co-ordinates of this point p\ :

i(2kt^ = 0- î^ktÛÔ(1)

As T^ and T^^, in addition to the co-ordinates of the un-

known point Pj, contain also the co-ordinates of the known

points p^ and p^, we may consider the two equations thus

established as being simply two equations between the six

co-ordinates of the three points iu question. These equations,

therefore, can be applied, not merely to determine the co-

ordinates of the point in the intermediate plane from those

* These lines are shewn canred in the figure, to indicate that the path

taken by a ray between two given points need not be simply the straight line

drawn between the two points, but a different Une determined by the re-

fractions or reflections which the ray may undergo: it may thus be either a

broken bne, made up of several straight lines, or (if the medium through

which it passes changes its character continuously instead of suddenly) a

continuous curve.



CONCENTEATION OF BAYS OF LIGHT AND HEAT. 303

of the two other points, but generally to determine any two

of the six co-ordinates from the other four.

Next let us assume that the points p^ a,ndp^ (Fig, 26),. in

which the ray cuts planes a and h, are

given, and that the point is yet un-

known in which it cuts plane c. This

point let us call p""^, to distinguish it

from other points in the same plane.

Take any point p^ in plane c, and con-

sider two auxiliairy rays, one of which

goes from p^ to p^, and the other from

Pi to p^. In Fig. 26 these are againshewn dotted, while the primary ray

is shewn full. Let T^^ and T^„ be the

times of passage of these auxiliary

rays. Then the value of the differ-

ence T^^ — Tj^ depends on the position

of p^ in plane c. Among the various

values obtained by giving to p„ various positions in the neigh-

bourhood of p\, the maximum must be that obtained by

making p^ coincide with p\. For in that case the ray passing

fromj?^ to p^ cuts the plane b in the given pointjp^, and is there-

fore made up of the ray which passes from p^ to p^ and of

that which passes from p^ to p^. Accordingly we may put

Hence the required difference is given in this special case by

T —T =Tas -'Sc -âS*

If on the other hand p^ does not coincide with p'^, then

the ray which passes from p^ to p^ does not coincide with

the two which pass from p^ to p^ and from p^ to p^ ;and

since the direct ray between p„ and p^ travels in the shortest

time, we must have

and therefore we have in general for the required difference

the inequality -

T — T <iT'an -^Sc^-^Bl'




This difference is thus generally smaller than in the

special case where p^ lies in the continuation of the ray

which passes from 'p^ to 'p^, and this special value of the

difference thus forms a maximum*. Hence wehave the

two following conditions

d^.'

dy,^>-

If we lastly assume that the points p^ and p^ in the

planes h and c are given beforehand, while the point in which

the ray cutsthe plane

ais still unknown, we obtain by an

exactly similar procedure the two following conditions

d \Tm ~ TJ) _ ^ _a[I^~- lî) _ »

f,„^

dx^'

dy.

In this way we arrive at three pairs of equations, each

of which serves to express the corresponding relation bet-êen

the three points in which a ray cuts the three planes a,h,c;so that if two of the points are given the third can be

found, or, more generally, if of the six co-ordinates of

the three points four are given the other two may be deter-

mined.

§ 6. Belation between Corresponding! Elements of Sur-

face.

We will now take the following case. Given on one

of the planes, say a, a point p^, and on another, say b, an

element of surface ds'^; then if rays pass from pa to the

different points of ds^, and if we suppose these rays produced

till they cut the third plane c, they will all cut that plane

in general within another indefinitely small element of

surface, which we will call dsg (Fig. 27). Let us now deter-

mine the relation between ds^ and dso.

* In KirchhofE's paper (p. 285) it ia stated of the quantity there con-

sidered, which is essentially the same as the difference here treated of, except

that it refers to four planes instead of three, that it must be a minimum.This may possibly be a printer's error, and in any case an interchange of max-imum and minimum in this place would have no further influence, because

the principle used in the calculations which follow, viz. , that the dierential

coefficient =:0i holds equally for a maximum or a minimum.



CONCENTRATION OF KAYS OF LIGHT AND HEAT. 305

Fig. 27,

In this case, of the six co-ordinates which relate to each

ray (viz. those of the three points

in which the ray cuts the three

planes) two, viz. x^ and y^, aregiven beforehand. If we nowtake any values we, please for

00^ and y^, the co-ordinates x^ and

y^ are in general thereby deter-

mined. Thus in this case wemay consider x„ and y^ as two

functions of x^ and y^. As the

form of the element ds,, may beany whatever, let it be a rect-

angle da?5, dy„, and let us find the

point in plane c correspondisng

to every point in the outline of this rectangle. We shall

then have on plane c an indefinitely small parallelogram

which forms the corresponding element of surface.

The magnitude of this parallelogram is determined asfollows. Let \ be the length of the side which corresponds

to the side dx„ of the rectangle in plane h, and let (Xa;„) and

(\y„) be the angles which this side makes with the axes

of co-ordinates. Then

\ cos (\a:.) = £^ dx^ ; \ cos l^y^) = ^dx^.

Again, let /* be the other side of the parallelogram, and let

(>*%„) and (jiy^ be the angles it makes with the axes. Then

we have

li cos {ûc,) = -£jdy,

; fi cos {fiyc) =^ '^3'^ •

Let (V) be the angle between the sides X and /*. Then we

have

cos (V) = cos {\x,) cos i/ix,) + cos {\y,) cos (fiy,)

'dx, dx dy, dy\ dx.dy^

\d£t dyt dx^ dyjyt dx^dyj X/*

C. 20



306 - ON THE MECHANICAL THEORY OP HEAT.

Now to determine the area cfo, of the parallelogram, we

may write

ds^ = \fi sin (\/i)

= \fijl— cos' (\/t)

= ^\y-cos'(X/i)\y.

Here we may substitute for cos (X/i) the expression just

given, and for X" and fi" the following expressions derived

from the above equations :

,_{fdfoX

Then several terms under the root caricel each other, and

the remainder form a square as follows :

V \dx^ dy^ dy^ dxj "

This quadratic equation can therefore be solved at once.

Butit must be observed that the' difference within the

brackets may be either positive or negative, and, as we

have only to do with the positive root, we will denote this

by putting the letters v.n. (valor numericus) before this

difference. We can then write

To ascertain how a;„ and y„ depend upon ir^ and y^, wemust apply one of the three pairs of equations in § 5. Wewill first choose equations (1). If we differentiate those

equations according to ss^ and ^j, remembering that'eacli of

the quantities denoted by T contains two of the three pairs

of co-ordinates x^, y^, x^, y^', x^, y„, as denoted. by the indices;

and if in differentiating we treat x, and .y.-as functions of



CONCENTRATION OF BAYS OF LIGHT AND HEAT. 307

ajj and y„ -whilst we take x^ and y^ as constant; then weobtain the following four equations

{dx,fX ? _L

dx^dx^ tfoj dx^dy,

+(fr..

+dx^dy^

d'(T^.+T,:)

(.dy>T+

dy^dx

dx„

dy^ dx^dy,

dXi

dy.

dx. dy^dy,

dy^dx^ dy^ dy^dy^

dyi

dxt

x^ =dy^

.(5).

If by help of these equations we determine the four

differential coefficients -j^°, z^' , -p

, ^ and subsititutetoj dy^ dx^ dy,,

the values thus found in equation (4), we obtain the required

relation between ds^ and ds„. To be able to write the result

more briefly, we will use the following symbols

A = v.n.(^dx^dx^

d^Z, d'Z,

dy^dy, dx^dy.

E= v.n.

.idx,r "" {dy,Y

dy.dxj W.

dx^dy^

•in

Then the required relation may be written as follows

dsg

E'A'

(8).

Again, if we suppose in like manner that a point p^ isgiven on plane c (Fig. 28), and find on plane a the element

da^, which corresponds to the given element ds^ on plane b,

then the result can be derived frofla that last given by simply

interchanging the indices a and c. If for brevity we write

a = vn fi^x^^Kdx^dXi dyjiy„ dxj,y^ dyM\

-)...(9).

20—2




then we have

ds, C.(10).

ib

Pig. 28.

Lastly, suppose a point p^ to be given on plane h (Fig. 29)

Fig. 29

and choose any element of surface ds^ on plane a. Let

us suppose that rays from different points of this element

pass through p^, and that they are produced to the plane c.

Then the magnitude of the element of surface dse, in which

all these rays meet plane c, is found, using the same symbols

as before, to be as follows

qA'

.(11).

From this we see that the.two corresponding elements in

this case bear exactly the same relation to each other, as the



CONCENTRATION OF BATS OF LIGHT AND HEAT. 309

two elements which are obtained when we have an element

rfSj given in plane b, and then, having assumed as the

origin of the rays a point first in plane a, and secondly

in plane c, determine in each case the element of surfacein the third plane corresponding to the element da^,

§ V. Fractions formed out of six qucmtities to express

the Relations between Corresponding Elements.

In the last section we have only employed the first of the

three pairs of equations in § 5. We can however employ the

two other pairs (2) and (3) in the same manner. Each pair

leads us to three quantities of the same kind as those already

denoted by A, G, and E. These quantities serve to express

the relations between the elements of surface. Of the nine

quantities thus obtained, hoTvever, there are four which are

equal to each other, whereby the actual number is reduced to

six. The expressions for these six are here placed together

for the sake of convenience, although three of them have been

already given.

'{dx^dx^ dyjiy^ dx^dij, dy„dxj

'\dx^dx^ dyjiy^ dxjiy^ dyJ^J

\dxjx^ dyj,y^ dxjdy^ dyjxj

n ^„ {^{T^-TJ)^^

d^{T^-T^) \d^{T.-TJ Y\

^~ l~~W^ (dyj^ L âdy. J I

^-^•î [dx:f "". {dy,r L dx,dy, J J

^ =^-°i {dx:f (dyj L dx^y, j]

(I-).

By help of these six quantities every relation between

two elements of surface can be expressed by three different

fractions, .as may be shewn in "tabular form as foUows







Given two elements ds^ and ds^ lA planes a and c (Fig.

30), we will first determine the heat

which ds^ sends to ds^. For this pur-

pose let us suppose the intermediateplane h to lie parallel to plane a at a

distance p, which is so small, that

the part which lies between these

two planes of any ray passing from

ds^ to ds^ may be considered as a

straight line, and the medium through

which it passes as homogeneous. Let

us now take any point in element

ds^, and consider the pencil of rays

which passes from this point to the

element ds^. This pencil will cut

plane h in an element ds^ whose magnitude is given by one

of the three fractions in the uppermost horizontal row of

equations (II.). Choosing the last of these we have the

equation

, Fig. 30.

^ds. (14).

The quantity G may in this case be brought into a

specially simple form, on account of the special position of

plane h. For this purpose let us follow Kirchhoff in choos-

ing the system of co-ordinates in h so as to correspond

exactly with that in the parallel plane a; i.e. let the origins

of both lie in a common perpendicular to the two planes

and let each axis of one system be parallel to the correspond-

ing axis of the other. Let r be the distance between two

points lying on the two planes, and having co-ordinates x^,

y^ and x^, y^ respectively. Then

r^Jp'+ix.-xS + iy-yJ' .(15).

Let us now suppose a single ray to pass from one of these

points to the other ; then, since its motion between the two

planes is supposed to be rectilinear, the length of its path

will be simply represented by r ; and if we denote by v^ its

velocity in the neighbourhood of plane a, which by the

assumption, will remain nearly constant between a and b,



312 ON THE MECHANICAL THEORY Ot HEAT.

then the time which the ray expends in the passage will be

given by the equation

"' v.-

The expression for C may therefore be written

1 / d'r ^^ cPr ^V^^

CPr \

~^'^' v^Kdx^dx^ dy^dy^ dx^dy^ dy^dxj'

Substituting for r its value as given by (15), we obtain

G =l-^^'? W

Hence equation (14) becomes

ds, = v:x-,Bds^ .(17).

r

If we denote by the angle which the indefinitely small

pencil of rays, which starts from a point on ds^, makes with

the normal at that point, then cos ^ = - ; and the last equa-

tion also takes the form

'"'""''^.Bds, (18).""'cos=^^

§ 9. Expressions for the quantities of Heat which ds„ and

ds„ radiate to each, other.

When the magnitude of the element of surface ds^ is

determined, the quantity of heat which ds^ sends to ds, can

be easily expressed.^

From every point of ds^ an indefinitely small pencil of

rays goes to.d&/,

andthe solid angle of the cone

madeby the

pencil from each of these points may be taken as the same.

The magnitude of this angle is determined by the magnitude

and position of that element ds^ in which the cone cuts

plane b. To express this angle geometrically, let us suppose

that a sphere of radius p is drawn round the point from

which the rays start, and that within this sphere we mayconsider the path of the rays as being rectilinear. If da is




the element of surface in -which this sphere is cut by the

pencil of rays, then —^ represents the angle of the cone. But

since the element ds^ is at the distance r from the vertex of

the cone, and since the normal to the surface at ds^, which

is parallel to the already mentioned normal to the surface at

ds^, forms with the indefinitely small cone of rays the

angle 6, we have the equation

da- _ cos X ds^ . .

— i \1"J'

If we substitute for «?«, its value from (18), we obtain

d^^^Bds^ (20).

We have now to determine the magnitude of that part of

the heat sent from ds^ which corresponds to this indefinitely

small cone; or in other words, how much heat ds^ sendsthrough the given element da- upon the spherical area. This

quantity of heat is proportional (1) to the magnitude of the

radiating element ds^, (2) to the angle of the cone, or to

-^, and (3), according to the well-known law of radiation, tor

the cosine of the angle 6, which the indefinitely small cone

makes with the normal. It may therefore be expressed by

e cos ^ -^ ds^,

P

where e is a factor depending on the temperature of the

surface. To determine this factor we have the condition

that the whole quantity of heat which tfe„ radiates, or which

it sends to the whole surface of any hemisphere above plane

a, must equal e^ds^ where e„ is the intensity of emission fromplane a at the position of ds^. Hence we have

?!'! cos 0da- = e^

The integration extends over the whole of the hemisphere,

and gives





CONCENTEATIOlJ OF EATS OF LIGHT AND HEAT, 315

depends on the nature of the medium, it becomes necessary

to consider this medium, and the method of determining its

influence.

If from the above two expressions we omit the factor

which is common to both, viz. — ds^ds^, we have the result

that the quantity of heat, which the element ds^ sends to the

element ds^, bears to that which ds^ sends to ds„ the ratio

ââ '• â"- If wê now assume that at equal temperatures

the radiation is always equal, even when the media in contact

with the two elements are- different, then for equal tempera-

tures we must put e^=e^', and the quantities of heat, which

the two elements radiate to each other, would then not be

equal, but would be in the ratio u/ : v^. It would follow

that two bodies which are in different media, e.g. one in

water and the other in air, would not tend to equalize

their temperatures by mutual radiation, but that one would

be able by radiation to raise the other to a higher temperature

than that which itself possesses.If on the other hand we maintain the universal correct-

ness of the fundamental principle- laid down by the author,

viz. that heat cannot of itself pass from a colder to a hotter

body, then we must consider the mutual radiations of two

perfectly black elements of equal temperature as being

themselves equal, and must therefore put

eav: = e,v:^ (21).

Hencee,, :c, ::^;/^^^/ (22).

Since the ratio of the velocities is the reciprocal of that

of the coefficients of refraction, which we m&y call n, and m„,

this proportion may be written

e.:c„ ::«/:< (23).

Hence the radiation of perfectly black bodies at equal

temperatures is different in different media, and varies in-

versely as the squares of the velocities in those media, or

directly as the squares of the coefficients of refraction. Thus

the radiation in water must bear to that in air the ratio

{^y : 1, nearly.



316 ON THE MECHANICAL THEORY OP HEAT. >

We have also to temember that in the heat radiated frofia

perfectly black bodies there are rays of very different wave-

lengths; and if we assume that the equality of mutual

radiation holds, not merely for the heat ais a whole, but also

for each wave-length. in particular, we must have for each

of these a proportion similar to (22) and (23) but in which

the quantities in the right-hand ratio have somewhat dif-

ferent values.

Lastly, instead of perfectly black bodies, let us consider

bodies in which the absorption of the rays received is partial

only. We must then introduce in the formula, in place of

the emission, a fraction having the emission as numerator andthe coefficient of absorption as denominator. For this fraction

we can obtain relations similar to those obtained for the

emission alone. This generalization, in which the influence

of the direction of the rays upon the emission and absorption

must also be taken into account, need not here be entered

upon.

IV. Determination of the mutual radiation of

TWO ELEMENTS OF SURFACE, IN THE CASE WHEN ONE

IS THE OPTICAL IMAGE OP THE OTHER,

§ 11. Relations between B, T>, F and E,

We have hitherto assumed that the planes a and c, so

far as we are concerned with them, give out their rays insuch a way, that one ray and one only, or at most a limited

quantity of individual rays, pa^ss from any point in the one

to any point in the other. We will now pass on to the case in

which this does not hold. The rays which diverge from points

in the one plane may be made to converge by reflections

or refractions, and to meet again on the other plane ; so that

for any point p^ on plane a there may be one or more points

or lines on plane c, in which an indefinitely large numberof the rays coming from p^ cut that plane, whilst other parts

of the same plane receive no rays whatever from p^. The

same of course holds of the rays which start from plane e

and arrive at plane a, since the rays passing to and fro be-

tween the same points describe the same paths.

Among the innumerable cases of this description we will,



CONCENTRATION OF RAYS OF LIGHT AND HEAT. 317

Kg. 31.

for the sake of clearness, consider

fixst the extreme case illustrated

in Fig. 31. In this case all the

rays sent out by J3„ within a cer-tain definite cone meet again in a

single point p^ of plane c. This

case occurs for example, when the

deflection of the rays is effected

by a lens, or by a spherical mirror,

or by any system of concentric

lenses or mirrors. We are here

supposed to neglect the spherical

and chromatic aberration, which

we have a right to do with regard

to the latter, inasmuch as we have confined ourselves to

homogeneous rays. Two points thus corresponding to each

other, as the points of starting and of re-union of the rays,

are called, as already mentioned, conjugate foci. For each

ray in such a case the co-ordinates «„, y„of

thepoint

p^ inwhich it strikes plane c, are determined by the co-ordinates

x^ y„ of the starting point p^. The other points on plane c

in the neighbourhood of p^ receive no rays from p^; for

there is no path to them which has the property that the

time in which the ray would traverse it is a minimum, as

compared with the time in which it would traverse any other

adjacent path. Hence the quantity T^ which expresses

this minimum time, can have a real value only for p^, andnot for any of the points round it. The differential coeffi-

cients of 2'^, in which the co-ordinates x^, y^ are assumed to

be constant and one of the co-ordinates x^ y^ to be variable,

(or conversely x^, y^ to be constant and one of the co-ordinates

x^ 2/o*o be variable) can thus have no real finite values.

It follows that of the six quantities A, B, G, D, E, F, which

have been determined by equations (I.), the three B,^ D, Fare not applicable to the. present case, inasmuch as they

contain differential coefficients of r„; whilst the three

others A, G, E contain only differential coefficients bf

T^ and T^„. Let us now assume that plane h is so chosen

that between it and the planes a and c, so far as we are

concerned with them, the radiation takes place on the

former system, so that one ray and one only, or at most a



318 OK THE MECHANICAL THEORY OF HEAT.

limited number of rays, pass from any point on plane b

to any point on plane a or c. Then for all the points with

which we are concerned, the quantities T^ and Tj„ and their

differential coefficients have real values not indefinitely large.

The quantities A, C and E are then as applicable in this case

as in the former.

One of these quantities, E, takes in this case a special

value, which may at once be found. For -the three points

in which the ray cuts the three planes a, h, «, the two equa^

tions given in (1) must hold, viz.

In the present case the position of the poirCt in which the

ray cuts plane h is not determined hj the position of the two

points p^ and p^, but plane h may be cut in all points of

a certain finite area. Hence the two equations above must

hold for all these points, and therefore the equations obtained

by differentiating these according toajj

and y^ must also hold,viz.

cP (?:.+ rj _.. d' (T,,+ TJ _ d' (T^+ TJ _ ^

da,:-"'

dx,dy,-"'

d^^-U...C^4;.

If we apply this to the equation determiningE in equa--,

tions (I.), we obtain

E= Q

(25).

The two other quantities A and C have in general finite

values, which differ in different circumstances, and which wemust now use in our further investigation, '

•

§ 12. Application of A and O to determine the Relation

bstween the Elements of Surface.

Let us suppose that the element ds^ on plane a has anoptical image ds^ on plane c, so that every point of ds^ has

a -point on ds^ as its conjugate focus, and vice vers^. Wehave now to enquire whether the quantities of heat, whichthese elements send to each other, when taken as elements

of the surfaces of two perfectly blaclc bodies of equal tem-peratures, are equal or not,.



CONCENTBATION OF RAYS OP LIGHT AND HEAT. 319^

First, to determine tlie position and magnitude of the

image ds„ corresponding to the given element ds^. Take

any point p^ on the intermediate plane h, and consider all

the rays starting from points on ds^, which pass throughp^. Each of these rays strikes plane c in the conjugate focus

to that from which it starts ; and therefore the element of

surface in which this pencil of rays cuts plane c is pre-i

cisely the optical image ds„ of the element ds^. Therefore,

to express the ratio between the ^reas of ds„ and ds^, wemay use one of the three fractions in the lowest row of

equations (II.)> which express the relation between the two

elements of surface, in which an indefinitely small pencil,

passing through a single point ^j of plane b, cuts the two

planes a and c. Of these three fractions the first is alone

applicable in this case, since the other two are undetermined.

We have thus the equation

1=^ ••• -f^^)-

. This equation is interesting also from an optical point

of view, as being the . most general equation which can be

given to determine the ratio between the area of an object

and that of its optical image. It should be remarked that

the intermediate plane h, to which the quantities A and Gare related, may be any whatever, and can therefore in any

particular case be chosen as is most convenient for calcula-

tion.

§ 13. Relation between the Quantities of Heat which

ds^ and dSj radiate to each other.

Having thus determined the element of surface ds^ which

forms the image of ds^, let us take on plane b, instead of a

single point, an element of surface ds^, and let us con-^

sider the rays which the two elements ds„ and ds^ send

through this element dsy All the rays, which start from

one point of ds^ and pass through ds^, unite again in on^

point of ds^ ; and thus all the rays which ds„ sends through?

dsj exactly strike the element ds,i and vice yers^. The

two quantities of heat which ds^ and ds„ send to ds^ are thus

the same as the quantities of heat. which ds^ and ds^ sea^




to each other through the intermediate element ds^. These

quantities of heat are therefore given at once by what has

gone before. Thus for the quantity of heat which ds^

sends to ds^, the sameexpression

willhold, as held in

§ 9 for the quantity of heat which ds^ sends to ds^, provided

we substitute G for B and ds^ for ds^. The expression thus

becomes

eji^^ — ds^ ds^.IT

Similarly for the quantity of heat which ds^ sends to ds^,

theexpression will

bethe

sameas that for the

quantityof

heat which ds^ sends to ds^, provided we substitute A for 5and dsj for ds^, or will be

e„«/ — dsjds^.IT

Kemembering that by equation (26)

Gds^= Ads^,

we see that these two expressions stand to each other in the

ratio e„w/ : e,v'.

We obtain precisely the same result, if we take any other

element ds,, on plane b, and consider the quantities of heat

which the two elements ds^ and ds^ send to each other

through this new element. These will , always be found to

stand to each other in the ratio e^t;/ : e„i;/. Since the quan-

tities of heat, which ds^ and ds„ send to each other on thewhole, are made up of those which they send to each other

through all the different elements of the intermediate plane,

the same ratio "must hold for the whole ; and we thus obtain

the final result, that the total quantities of heat which ds^ and

ds^ radiate to each other, stand in the ratio ej!^ : e^vf.

This is the same relation as was found in sections 8 and 9

for the case where there is no concentration of rays ; it thus

follows that the concentration of rays, however much it alters

the absolute magnitudes of the quantities of heat which two

elements radiate to each other, leaves the ratio between themexactly the same.

It was shewn in § 10, that if in the case of ordinary un-

concentrated radiation the principle holda that heat cannot

pass from a colder to a hotter body, then the radiation must



CONCENTRATIOK OF RAYS OF LIGHT AND HEAT. 321

differ in different media, and must beûch that for perfectly

black bodies of equal temperatures

If this equation is satisfied, then in the present ease also,

where the elements ds^ and ds^ are images one of the other,

the quantities of heat which they mutually radiate must beequal; and therefore, in spite of the concentration of the rays,

one element cannot raise the other to a higher temperature

than its own.

V. Relation between the increment of area and theRATIO or THE TWO SOLID ANGLES OP AN ELEMENTARYPENCIL OF RAYS.

§ 14. Statement of the Proportionsfor this Case,

As an immediate result of the foregoing we may here

develope a proportion, which appears to have some general

interest, inasmuch as it illustrates a peculiar difference in the

behaviour of a pencil of rays in the case of an object and of

its image. This difference' must always exist and have a de-

terminate value, when the object and the image have differ-

ent areas.

Consider an indefinitely small pencil of rays, which starts

from a point on ds^, passes through the element ds^ on the

intermediate, plane, and then unites again in a point on cfo„.

We may compare the divergence of the rays at their starting

point with the convergence of the same at their point of re-

union. This divergence and convergence (or, to use the

ordinary phrase, the solid angles ofthe indefinitely small cones,

which the pencil forms at its points of starting and re-union)

are given directly by the same method which we have used

in§ 9, d,g follows:—•'

•

Suppose that round each of the points there is described

a sphere of so small a radius, that we may consider the i ays'

as going in straight lines as far as the surface of this sphere:

and then consider the element of surface in which the pencil

bif rays cuts the sphere. Let do- be this element and let p be

the radius of the sphere; then the angle of the indefinitely

small cone, which contains the rays so far as they are recti-

c. 21




linear, will' be expressed by —^ . This fraction we have de-

termined for a similax case in § 9 by equation (20), and in

the expression there given we have only to alter the letters

in order to transform it into the expression for the present

case. In order to express the angle of the cone for that

starting; point of the rays which lies4n plîe a, we have only

to substitute in that expression ds^ for ds^ and G for B. In

addition,, ;the symbol ^,. which expresses the angle between

the pencil of rays and the normal to the surface of ds^, maybe changed to 6^ , so as to express more clearly that it relates

to plane a; and for the same reason the suffix a may be

added to — j, ., Thus we obtain

^4]=^Gds, (27),

To obtain the other equation, which gives the angle of the

cone at the point of re-union on plane c, we have only to

change the S'uffia^ a into c throughout, and also to substitute

A for G. Thus we have ,

^°'^"" Ads^ (28).

!da

./)V„ cos^,

From these two equations we obtain the proportion,

cos 5„ (d<T\ cos B /da-

since by equation (26) Gds^= Ads^.,

If we substitute for the velocity the coefficient of refraction,

this proportion becomes

n^ cos e„ [-?) . : K cos 0, (-^ j:: ds, : ds„ (30).

Here the ratio on the right-hand side is the ratio between

the. area of an element of surface of the image, and the area of

the. corresponding element of the object, or in short the propor-

tionate increment of area ; and we thus obtain a simple rela-

tion between this increment and the ratio of the angles of the

cones made, by an elementary pencil of rays. It is easily seen



CONCENTEATION OF EAYS OF LIGHT AND HEAT. 323

that it is not necessary for the truth of this proportion that the

rays should finally converge, and meet at one point, but they

may also be divergent, in which case their directions meet in

one poiiit

when produced backwards, and form what is calledin optics a virtual image.

If we take the special case in which the medium at the

point of starting and of re-union is the same (e.g. where

the rays issue from an object which is in air, and, after cer-

tain refractions or reflections, form an image which really

or virtually is also in air), then v^ = v„ and w^ = «„ ; whence

we have

cos0„f-^j : cos0j-?j :: ds^ : ds^.

If we add the further condition, that the pencil of rays

makes equal angles with the two elements of surface (e.g.

that both are right angles), then we have

/dcr\. fdi

{?)At);-'''''

In this case the angles of the cones formed by the pencil

of rays at the object and at the image stand simply in the

inverse ratio of the areas of the corresponding elements of

object and image.

In the valuable demonstration which Helmholtz has givenin his "Physiological Optics"* of the laws of refraction in

the case of spherical surfaces, he seeks to connect with these

the case of the refractions which take place in the eye ; and

he' finds in page 50, and extends further in page 54, an equa-

tion which expresses- the relation between the size of the

image and the convergence of the rays, for the case in which

the change of direction of the rays takes place by refraction

or by reflection at the surface of co-axial spheres, and in which

the rays are approximately perpendicular to the planes which

contain the object and the image. The author however be-

lieves that the relation' has not before been given in its

general form, as in proportions (29) and (30).

• Karsten's Universal Eneyelopedia of Physics.

21—2



32-l! ON THE MECHANICAL THEORY OF HEAT.

VI. General determination of the mutual radiation

BETWEEN TWO SURFACES, IN THE CASE WHERE' ANY

CONCENTRATION WHATEVER MAY TAKE PLACE.

§ 15. General View of the Concentration of Rays.

We must now extend our investigation so as to embrace

not only the extreme case, in which all the rays which issue

from a point on plane a within a certain finite cone unite

again in one point forming a conjugate focus on plane c, but

also every conceivable case of the concentration of rays.

To obtain a closer view of the phenomena of concentration,

we may use the following definition. If rays issue from any

point p^ and fall on plane c, and if these rays when close to

that plane have such directions that on one part of the plane

the density of the impinging' rays is indefinitely great com-

pared to the mean density, then at this part there is concen-

tration of the rays which issue from p^. With this definition

we may easily treat mathematically the case of concentration.

Between point p^ and plane c take any intermediate plane h,

which is so placed that there is no concentration in it of the

rays issuing from p^ ; and also that its relation to plane c is

such, so far as we are concerned, that the pencils of rays issu-

ing from points on one of those planes suffer no concentration

on the other. Now consider an indefinitely, small pencil,

which starts from p^ and cuts the planes h and c. Let uscompare the areas of the elements ds^ and ds^ in which these

planes are cut. If ds^ vanishes in comparison to ds^, so that

we may put

|-° (5').

this is a sign that there is a concentration of rays, in the

sense defined above, at plane c.

Let us now return to equations (II.) of § 7. The equations

in the first horizontal row are those that refer to the present

case:and of the three fractions, which there represent the

ratio of the elements of surface, the first applies to our case,

because under the assumption made as to the position of the

intermediate plane we maydetermine A and E in the ordi-



CONCENTRATION OF BAYS OF LIGET AND HEAT. 325

nary manner. "We have thus the equation

ds^ A'

This fraction can only equal zero if the numerator E is

zero, since under the assumption made as to the position of

the intermediate planes the denominator A cannot be indefi-

nitely large. We have then as a mathematical criterion,

whether the rays issuing from p^ suffer concentration at plane

c or not, the condition

E=Q (32),.

which must be fulfilled where there is concentration.

Now assume conversely that on plane c a point p^ is given,

and that we have to decide whether the rays issuing from

this point suffer concentration at any part of plane a. Then

dswe have in the same way the condition t-" = ; and since by

(II.) -T^= ^,

we arrive at the same final condition

i;=o.

It is in fact easy to see that when the rays issuing from a

point on plane a suffer concentration at plane c, then con-

versely the rays issuing from the latter point must suffer con-

centration on plane a.

Since equations (13) express the relations which hold be-

tween the six quantities J, B, G, 3, E, F, we may apply

those equations to ascertain what becomes of B, D and F,

in the case where E = 0, whilst A and have finite values.

By those equations

^=§-' ^=?' ^=i"••••(^^)-

Hence it follows thatall

threequantities

mustin the present

case be indefinitely great.

§ 16. Mutual Radiation of an Element of Surface and

of a Finite Surface, through an Element of an Intermediate

Plane.

We will now attempt so to determine the ratio between

the quantities of heat which two surfaces radiate to each




other, that the result must hold in all cases, independently

of the question whether there is any concentration of heai

or no.

For greater generality we will substitute for the planes aand c, as hitherto considered, two surfaces of any kind, which

we may call s„ and s„. Between them let us take any third

surface Sj, which need only fulfil the condition that the rays

which pass from s„ to s„, or vice versS,, suffer no Qoncentration

in Sj. Now choose in s„ any element ds^, and in s^ an ele-

ment dsj, so situated that the rays, which pass through it

from ds^ will when produced strike the surface s„. Then we

will first determine how much heat ds^ sends through ds^ io

the surface s„, and how much heat it receives back from s„

through the same element of the intermediate, plane. To

ascertain the first mentioned quantity of heat, we have only

to determine how much heat ds^ sends to ds^, since, by our

assumption as to the position of ds^, all this heat after pass-

ing dsj must strike the surface s„. This quantity of heat may

beexpressed at once by means of our previous formulae.

Suppose a tangent plane to be drawn to s^ at a point of the

element ds^, and similarly a tangent plane Sj at a point oids^;

and consider the given elements of surface as elements of

these planes. If in these tangent planes we take two sys-

tems of co-ordinates a;„, y^, and x^, y„, and if we form the

quantity G by means of the third of equations (I.), then the

required quantity of heat, which ds^ sends through ds^ to the

surface s^, is given by the expressions

ej]^ — ds^dSi.IT

Next with regard to the quantity of heat which ds^ re-

ceives through dsj from the surface s„, the relations of the

points in s^, from which these rays issue, are not in general

so simple as that which holds in the special case, where ds,

has an optical image ds„ lying on s„, and therefore is also

itself the optical image of ds„. If we choose a known point

p^ on the intermediate element ds^, and consider the rays

which pass through this point from all points of ds^, we have

an indefinitely small pencil of rays, cutting s^ in a certain

element of surface. It is this element which sends rays to

ds^ through the selected point p^. But if we now choose



CONCENTRATION OF RATS OF LIGHT AND HEAT, 327

.another point of ds^ as the vertex of the pencil, we arrive

at a somewhat different element on the surface s^. Thus. the rays, which ds^ receives from s^ through different points

on ds^, do not all issue from one and the same eleinent

of s,.

Since however the area of ds^ may be any whatever,

nothing hinders us from supposing it so small, that it is an

indefinitely small quantity of a higher order than ds^ In

this case, if the vertex of the pencil changes its position

within dsf, then the element of «„ which corresponds to ds^

will change its position through a distance so small that

in comparison with the dimensions of the element it is

indefinitely small and may be neglected. Hence in this

case the 'element ds^, which we obtain when we choose any

point whatever p^ on element ds„i and make it the vertex of

the pencil of rays issuing from,ds„, may be considered as the

part of ds^ which exchanges rays with ds^ through ids^. The

area of this element ds^ is easily found from what precedes.

Let us suppose as before that a tangent plane to the surface s,

is drawn at p^, and that tangent planes to the surfaces s„ and

s„ are drawn at points on the elements ds^ and ds^ respectively;

and let us consider the two latter elements as elements of

the tangent planes. Take systems of co-ordinates on these

three tangent planes, and form the quantities A and C, as

given by the first and third of equations (I.). Then by

equation (II.) we have

' ds,= -^d8^.

The quantity of heat which <i5„, sends to ds„ and which,

as mentioned above, may be considered as the quantity which

ds^ receives from the surface s, through ds„ is expressed, by

IT

or, substituting for ds^ its value given above.



S28 . ON THE MECHANICAL THEOKT OF HEAT.

If we Compare this expression with that found above,

which expresses the quantity of heat sent by ds^ through ds,

to s„, we see that the two stand to each other in the ratio

e^v^ : efi^. If we now suppose that «„ and s„ are the sur-

faces of two perfectly black bodies of equal temperature, and

make for such surfaces the assumption (which we have al-

ready seen to be necessary in the case of radiation without

concentration), that the two products e„v/ and e^^ are equal,

then the quantities of heat given by the two expressions above

are also equal.

§ 17. Mutucd Radiation of Entire Surfaces.

If on the intermediate surface s^ we take, instead of the

element hitherto considered, another element which is also

supposed to be an indefinitely small quantity of a higher

order, then the element of s^, which exchanges rays with ds^

through this element of s^, has a different position from that

in the last case ; but the two quantities of heat thus ex-

changed are again equal to each other;

and the same holdsof all other elements of the intermediate surface.

To obtain the quantity of heat which ds^ sends to s„, not

through a single element of the intermediate surface, but as

a whole, and similarly the quantity of heat which as a whole

it receives from s„, we must integrate the twp expressions

found above over the surface s„ so far as this surface is cut

by the rays which pass from ds^ to s„, and vice verset. It is

evident that if for each element of surface ds^ the twodifferentials are equal, then the whole integrals must also

be equal.

Lastly, to find the quantities of heat, which the whole

surface s^ exchanges with s„, we must integrate both these

expressions over the surface *,. This process again will not

disturb the equality, which exists for each of the separate

elements ds^.

Thus the principle previously discovered in a special case,

viz. that two perfectly bla,ck bodies of equal temperatures ex-

change equal quantities of heat with, each other, so far as the

equation ejD^ —^^^ holds for them, appears also as the result

of an investigation, which in no way depends upon whether

the rays issuing from s, suffer a concentration at s„ and vice

vers^, or not; since the only condition was, that the rays



CONCENTRATION OF EATS OF LIGHT AND HEAT. 329

issuing from s^ and s„ suffer no concentration at the inter-

inediate surface Sj, a condition which may be always fulfilled,

since this surface may be chosen at pleasure.

It follows further from this result that, if a given blackbody exchanges heat not only with one but with any numberof black bodies, it receives from all of them exactly the samequantity of heat as it sends to them.

§ 18. Consideration of Various Collateral Circumstances.

The previous investigation has beenmade

throughout

under the assumption that any reflections and refractions

take place without loss, or that there is no absorption of

heat. We can, however, easily go on to shew that the results

are not altered, if this condition is dropped. For consider

any one of the different processes, by which a ray may be

weakened on its way from one body to another ; say that at

a place where the ray cuts the boundary of two media, one

part passes into the further medium by refraction, and theother is reflected. Then whether we consider the one part

or the other as the continuation of the original ray, we have

in both cases a weakened ray to deal with. The same holds

if a ray be partially absorbed by entering a medium. But

in each of these cases we have the law that two rays which

traverse the same path in opposite directions are weakened

in equal proportion. The quantities of heat, which two

bodies mutually send to each other, are therefore weakened

by such processes to the same extent ; so that, if they would

be equal without such weakening, they will also be equal

when thus weakened. Another circumstance may be con-

sidered in connection with the processes above mentioned,

viz. that a body may receive from the same direction rays

which proceed from different bodies. For example, a body A

may receive from a point, which lies on the bounding surfaceof two media, two rays coinciding in direction, but issuing

from two different bodies B and 0. One of these may come

from the bounding medium and be refracted at the point,

whilst the other is already in the bounded medium, and is

reflected at the point. In this case, however, the two rays are

weakened by refraction and reflection in such a way, that, if

both were before of equal intensity, their sum afterwards is




of the same intensity as either one of them had beforehand.

Now suppose a ray of the same intensity to. issue from the

body A in the opposite direction,, this will be divided, at the

same point, into two parts, of which one enters the boundingmedium, and passes forward to the body B, while the other

is reflected and passes to the body C. The two parts which

thus reach B and fromA are exactly as great as those which

A receives from B and C. The body A thus stands to each

of the bodies B and G in .such a relation, that, assuming

equal temperatures, it exchanges with them equal quantities

of heat. The equality of the modifications which two rays

undergo, when passing in opposite directions in any path

whatever, must produce the same result in all other cases

however complicated.

Again i^ ipstead of perfectly bla>ek bodies, we consider

such as only partially absorb the rays falling on them; or if

instead of homogeneous heat we consider heat. which con-

tains systems of waves ©f different lengths ; or lastly, if

instead of taking all the rays as unpolarized weinclude the

phenomena of polarization; still in all these cases we have to

do only with facts, which hold equally for the heat sent out

by any one body,.and for that which it receives from other

bodies. It is not necessary to consider these facts more

closely, since they also take place with ordinary radiation

without concentration, and the object of the present investi-

gation was only to consider the special actions which might

possibly be produced by concentration of the rays.

§ 19. Sumrmiry of Restdts,

The main results of this investigation may be briefly

stated as follows

(1) In order to bring the action of ordinary radiation,

without concentration, into accordance with the fundamental

principle, that heat cannot of itself pass from a colder to a

hotter body, it is necessary to assume that the intensity of

emission from a body depends not only on its own composi-

tion and temperature, but also on the nature of the sur-

rounding medium; the relation being such, that the in-

tensities of emission in different media stand in "the inverse

ratio of the squares of the velocities of radiation in the



CONCENTRATION OF RAYS OF LIGHT AND HEAT. 331

media, or in the direct ratio of the squares of the coefScients

of refraction.

(2) IF this assumption as to the influence of the sur-

rounding media is correct, the above fundamental principle

is not only fulfilled in the case of radiation without concen-

tration, but must also hold good when the rays are concen-

trated in any way whatever by reflections or refractions

since this concentration may indeed change the absolute

magnitudes of the quantities of heat, which two bodies

radiate to each other, but not the ratio between these

quantities.



CHAPTER XIII.

DISCUSSIONS ON THE MECHANICAL THEOET OF HEAT

AS HERE DEVELOPED, AND ON ITS FOUNDATIONS.

§ 1. Different Views as to the Relation between Heat and

Work.

The papers of the author on the Mechanical Theory of

Heat, as reproduced in all essential particulars in this volume,have frequently met with opposition; and it may be desirahle

to give here some of the discussions which have taken place

on the question, since many points are raised in them, on

which the reader may even yet be in doubt. The removal

of these doubts may be facilitated by his learning what has

been already written upon these points.

As already mentioned in Chapter III., the first important

attempt to reduce to general principles the action of heat in

doing work was made by Carnot. He started from the

assumption that the total quantity of heat existing was in-

variable, and then supposed that the falling of heat from a

higher to a lower temperature brought about mechanical work,

in the same way as the falling of water from a higher to a

lower level. But simultaneously with this conception the

view gained ground thatheat

is

a mode of motion, and that inproducing work heat is expended. This view was set forth

at intervals from the end of the last century by individual

writers, such as Rumford, Davy, and Seguin*; but it was not.

tin 1840 that the law corresponding to this view, viz. that of

* In a paper by Mohr, publiBhed 1837, heat is in some places called a

molion, in otfieis a force.



, DISCUSSIONS ON THE THEORY, 333

tVie equivalence of neat and work, was definitely laid downby Mayer and Joule, and proved by the latter to be correct

by means of numerous and brilliant experiments. Soon

afterwards the general principle of the conservation of energywas laid down by Mayer* and Helmholtzf (by the latter in a

specially clear and convincing manner), and was applied to

various natural forces.

A fresh starting point was thus given to researches on

the science of heat ; but in carrying these out great difficul-

ties presented themselves, as was natural with so widely

extended a theory, which was intertwined with all branches

of natural science, and influenced the whole range of physi-

cal thought. In addition, the wide acceptation which Car-

net's treatment of the mechanical action of heat had won for

itself, especially after being brought by Clapeyron into an

elegant analytical form, was unfavourable for tlie reception

of the new theory. It was believed that there was no alter-

native but either to hold to the theory of Carnot, and reject

the new view according to which heat must be expendedto

produce work, or conversely to accept the new view and

reject Carnot's theory.

§ 2. Papers on the Subject hy Thomson and the Author.

A very definite utterance on the then position of the

question was given by the celebrated English physicist, now

Sir William Thomson, in an interesting paper which he pub-

lished in 1849 (when most of the above-mentioned researches

of Joule had already appeared and were known to him),

under the title, "An Account of Carnot's Theory of the

Motive-Power of Heat, with numerical results deduced from

Kegnault's ex;periments on steamJ."

He still maintains the

position of Carnot, that heat may do work without any change

in the quantity of heat taking place. He however points

out a difficulty in this view, and goes on to say, p. 545:

"It

might appear that this difficulty might be wholly removed, if

we gave up Carnot's fundamental axiom, a view which has

been strongly urged by Mr Joule," He adds,, <' but if we do

* Die organiscke Bewegung in ihrem Zusammenhange mit dem Stoff^

weeheel. Heilbronn, 1845.

+ Ueber die Erhaltung der Kraft, 1837.

T Trans. Royal Soc, of Edin., Vol. xvi. p. 541,




this we stumble over innumerable other difficulties, which

are insuperable without the aid of further experimental

researches, and without a complete reconstruction of the

Theory of Heat. It is in fact experiment to which wemust look, eitber for a confirmation of Carnot's axiom, and a

clearing up of the difiSculty whicb we have noticed, or for a

cornpletely new foundation for the Theory of Heat."

At the time when this paper appeared the author was

writing his first paper " On the Mechanical Theory of Heat,"

which was brought before the Berlin Academy in 1850, and

printed in the March and April numbers of Poggendorff's

Annalen. In this paper he attempted to begin the recon-

struction of the theory, without waiting for further experi-

ments; and he succeeded, he believes, in overcoming the

difficulties mentioned by Thomson, so far at least as to leave

the way plain for any further researches of this character.

He there pointed out the way in which the fundamental

conception, and the whole mathematical treatment of heat,,

must be altered, if We accepted the principle of the equiva^'

lence of heat and work ; and he further shewed that it was

not needful wholly to reject the theory of Carnot, but that-

we might adopt a principle, based on a different foundation

from Carnot's, but differing only slightly in form, which

might be combined with the principle of the equivalence of

heat and work, to form with it the ground-work of the new

theory. This theory he then developed for the special cases

of perfect gases and saturated vapour, and thereby obtained;a series of equations, which have been universally employed

in the form there given, and which will be found in Chap'

ters II. and VI. of this volume.

§ 3. On Rcunhine's Paper and Thomson's Second Paper.

In the same month (February, 1850) in which the

author's papej* was read before the Berlin Academy, a valu-

able paper by Eankine was read before the Royal Society of

Edinburgh, and was afterwards published in their Trans-

actions (;VoL 20, p. 147)*.

Rankine there proposes the hypothesis, that heat consists

in a rotary motion of the molecules ; and thence deduces with

* In 1854 it was reprinted -with some alterations, in the Fhil. Mag,,

Series 4, Yol. vn. pp. 1, HI, 172.



DISCtrSSIONS ON THE THKORT. 335

much skill a series of principles on the action of heat, which

agree with those deduced by the author from the first main

principle of the mechanical theory*. The second main prin-

ciple was not touched by Rankine in this paper, but wastreated in a second paper, which was brought before the

Eoyal Society of Edinburgh a year afterwards (April, 1851).

In this he remarka-f- that be had at first felt doubtful of the

correctness of the mode erf reasoning by which the author

had arrived at this second principle ; but that having com-

municated his doubts to Sir W, Thomson, he was induced

by him to investigate the subject more closely. Ke then

found that it ought not to be treated as an independent

principle in the theory of heat, but might be deduced from

the equations, which he had given in the first section of his

former paper. He proceeds to give this new proof of the

principle, which however, as will be shewn further on, is in

opposition, , for certain very important, cases, with his ownviews, as elsewhere expressed.

This paper of 1851 Rankine added as a fifth section to

his. former paper on account of the connection between them.

Thence has arisen with some authors the mistaken idea that

this new paper was actually part of the former one, and that

Rankine had therefore given a proof of the second main

principle at the same time as the author. From the fore-

going it will be seen that his proof (waiving the question

how far it is satisfactory) appeared a year later than the

author's.

In March of the same year, 1851, a second paper by Sir

W. Thomson on the Theory of Heat was laid before the

Royal Society of Edinburghf.

In this paper he abandons his

former position with regard to Carnot's theory and assents to

the author's exposition of the second main principle. He

then extends the treatenent of the subject. For whilst the

author had confined himself in the mathematical treatment

of the question to the case of gases, of vapours, and of the

process of evaporation, and had only added that it was easy

to see how to make similar applications of the theory.to other

• Edinb. Trans., Yol.,xx. p. 205; Phil. Mag., Series i, Vol. vii. p. 2i9.

t Phil. Mag., Vol. vii. p. 250.

j Edin. Trans., Vol. xx. p. 261 ; Phil. Mag., Series 4, Vol. iv. pp. 8, 105,

168.



330. ON THE MECHANICAL THEORY OF HEAT.

cdses, Thomson developes.a series ofgeneral equations, whieh

are independent of the body's condition of aggregation, and

only then passes on to more special applications.

On one point this second paper still falls short of the

author's. For here also Thomson holds fast by the law of

Mariotte and Gay-Lussac in the case of saturated vapour,

and hesitates to accept an hypothesis with respect to perma-

nent gases, which the author had made use of in his investi-

gation (see Chapter II., § 2, of this work). On this he

remarks* :" I cannot see that any hypothesis, such as that

adopted by Clausius fundamentally in his investigations on

this subject, and leading, as he shews, to determinations ofthe densities of saturated steam at different temperatures,

which indicate enormous deviations from the gaseous laws of

variation with temperature and pressure, is more probable,

or is probably nearer the truth, than that the density of sa-

turated steam does follow these laws, as it is usually assumed

to do. In the present state of science it would perhaps be

wrong to say that either hypothesis is more probable than

the other." Some years later, after he had proved by his

joint experiments with Joule that this hypothesis is correct

within the limits assigned by the author, hfe used the same

method as the author to determine the density of saturated

vapour"I".

Rankine and Thomson, so far as the author knows, have

always recognized most frankly the position here assigned to

thefirst

labours of themselves and the author on the me-chanical theory of heat. Thomson remarks in his paper

J:" The whole theory of the moving force of heat rests on the

two following principles, which are respectively due to Joule

and to Carnot and Clausius." Similarly he introduces the

second main principle as follows: "Prop, II. (Carnot and

Clausius)." He then proceeds to give a proof discovered by

himself, and goes on§: "It is with no wish to claim priority

that I make these statements, as the merit of first establish-

ing the proposition on correct principles is entirely due to

Clausius, who published his demonstration of it in the month

• Edin. Trans., Vol. xx. p. 277; Phil. Mag., Yol. iv. p. 111.

t Phil. Trans., 1854, p. 321.

$ Mdin. Trans., Vol. xx. p. 264 ; Phil. Mag., Vol. iv; p. 11;

§ Edin. Trans., Vol. xx. p. 266 ; Phil, Mag., Vol. iv. pp. 14, 242. '

'



DISCUSSIONS ON THE THEOET. 337'

of May last year, ia the second part of his paper on the

Motive Power of Heat."

§4, Holtzmann's objecticms.

From other quarters repeated and in some cases violent

objections were raised to the author's first paper, to which, in

the same and following years, a series of other papers, serving

to complete the theory, were added. The earliest of these

objections came from Holtzmann, who had published in 1845

a short pamphlet* on the subject. In this it would at first

appear as if he wished to treat the question from the point

of view, that for the generation of work there was necessaiy

not merely a change in the distribution of heat, but also an

actual destruction of it, and that conversely by destroying

work heat could be again generated. He remarks (p. 7):

" The action of the heat which has passed to the gas is thus

either a raising of temperature, combined with an increase of

the elastic force, or a certain quantity of mechanical work, or

amixture of the

two; and a certain quantity of mechanical

work is equivalent to the rise in temperature,. Heat can only

be measured by its effects; of the two above-named effects

mechanical work is especially adapted for measurement, and it

will be chosen accordingly for the purpose. I call a unit of

heat that amount of heat, which by its entrance into a gas

can perform the mechanical work a, or, using definite mea-

sures, which can raise a kilograms to a height of 1 metre."

Further on (p. 12) he determines the numerical value of the

constant a by the method previously used by Mayer, and

explained in Chapter II., § 5; the number thus obtained cor-

responds perfectly with the mechanical equivalent of heat, as

determined by Joule by various other methods. In extend-

ing his theory however, i.e. in developing the equations by

which his conclusions are arrived at, he follows the same

method as Clapeyron; so that he tacitly retains the assump-

tion that the total quantity of heat which exists is invariable;

and therefore that the quantity of heat which a body takes

in, while it passes from a given initial condition to its

present condition, must be expressible as a function of the

variables, which determine that condition.

* Ueber die Warme und Etasticitdt der Gase und Dampfe; von C.

Holtzmann. Mannheim, 1845 ; also Pogg. Am., Vol. lxxii a.

Q. 22




In the author's first paper the inconsequence of this

method was pointed out, and the question treated in another

way.; on which Holtzmann wrote an article*, in which he

endeavoured to shew that this method of treatment, andspecially the assumption that heat was expended in pro-

ducing work, was inadmissible. The first objection which

he raised was of a mathematical character. He carried out

an investigation similar to that in the author's paper, in

order first to determine the excess of the heat which a body

takes in over that which it gives out, during a simple cyclical

process consisting of indefinitely small variations, and secondly

to compare this excess with the work done. But iii such a

process both the work done and the excess of heat must be

indefinitely small quantities of the second order ; and there-

fore in the whole investigation, care must be taken that all

quantities of the second order, which do not cancel each

other, shall be taken into account. This Holtzmann neg-

lected to do ; and he was thus led to a final equation, which

contained a self-contradiction, and in which he therefore

imagined that he had found a proof of the inadmissibility of

the whole method. This objection was easily disposed of

by the author in his reply.

He further brought forward as an obstacle to the theory,

that, according to the formulae given, the specific heat of a

perfect gas must be independent of its pressure, whereas the

experiments of Suermann, and also those of De la Roche

and B^rard, shewed that the specific heat of gases increasedas the pressure diminished. On this conflict between his

own theory and the experiments which were then known

and supposed to be correct, the author remarked in his reply

as follows : "On this point I must first point out that, even

if these observations are perfectly correct, they yet say nothing

against the fundamental principle of the equivalence of heat

and work, but only against the approximate assumption

which I have made, viz. that a permanent gas, if it expand

at constant temperature, absorbs only so much heat, as is

required for the external work which it thus performs. Butbesides it is sufficiently known how unreliable are in general

the determinations of the specific heats of gases ; and all the

* Pogg. Ann., Vol. lxxxii. p. 445.






the differential equations there formed, which, though not

generally int^grable, become so as soon as one further rela-

tion is assumed to exist among the variables. In spite of all

•whichthe author has said, he has throughout treated the

quantities to which these equations refer, viz. the quantities

of heat taken in by a body in passing from a given initial

condition to its present condition, as mere functions of the

variables which determine the condition of the body. After

quoting the author's equation for gases, viz.

dt\dv) dvKdt) y'^ ''

•where A is the heat-equivalent of the unit of work, i.e. the

reciprocal of E, he remarks, page 243: "In equation (1)

( -^1 and (-5^) are fully determined as the differential co-

efficients of a known function of v and t, viz- Q, taken ac-

cording to V and t respectively as sole variables ; and in

whatever •way this function may be formed, aud whatever

relation may be supposed to exist between v and t, the right

side of the equation must always equal zero."

This incorrect conception, thus formed by a professed

mathematician, convinced the author that the meaning and

treatment of this kind of differential equation, although long

before established by Monge, was not so generally known as

he had supposed; accordingly in his reply*, after a brief

notice of some other points raised by Decher, he treated the

subject more fully, giving a mathematical explanation, which

seemed to him sufficient to obviate any such misunderstand-

ings in future. This was afterwards prefixed to the collection

of the author's papers as a mathematical introduction ; and

the essential part of it has been imported into the mathe-

matical introduction to the present work.

§ 6. Fundamental Principle on which the Author^s Proof

of the Second Main Priticiple rests.

The more recent objections to the author's theory, andthe departure from his views in more recent treatises, chiefly

• Dinglei's PolyUchnucher Journal, Vol. ci. p. 29,





Sii ON THE MECHAN'ICAL THEORY OF HEAT.

arises, whether it is not possible to concentrate the rays of

heat artificially by means of mirrors or burning-glasses, so as

to produce a higher temperature than that of the radiating

, bodies, and thus to effect the passage of the heat into ahotter body. The author has, therefore, thought it necessary

to treat this question in a special paper, the contents of

which are given in Chapter XII. Matters are still more

complicated in cases when heat is transformed into work, and

vice versi, whether this be by effects such as .those of

friction, resistance of the air, and electrical resistances, or

whether by the fact that one or more bodies suffer such

changes of condition, as are connected partly with positive

and partly with negative work, both internal and external.

For by such changes heat, to use the common expression,

becomes latent or free, as the case maybe; and this heat the

variable bodies may draw from or impart to other bodies of

different temperatures.

If for all such cases, however complicated the processes

may be, it is maintained that without some other permanent

change, which may be looked upon as a compensation, heat

can never pass from a colder to a hotter body, it would seem

that this principle ought not to be treated as one altogether

Self-evident, but rather as a newly-propounded fundamental

principle, on whose acceptance or non-acceptance the validity

of the proof depends.

§ 8. Zeuner's later Treatment of the Subject.

The mode of expression employed by Zeuner was criticized

by the author on the grounds stated in the last section, in a

paper published in 1863. In the second edition of his

book, published in 1866, Zeuner has therefore struck out

another way of proving the second main principle. Assum-

ing the condition of the body to be determined by the

pressure p and volume v, he forms, for the quantity of heatd Q, taken in by the body during an indefinitely small varia-

tion, the differential equation

dQ= A{Xdp+7dv)....(2),

where X and Y are functions of p and v, and A is the heat-

equivalent of work. This equation, as is well known, cannot



DISCUSSIONS ON THE THEOET. 343

be integrated so long as p and v are independent variables.

He then proceeds (p. 41)

"But let /S be a new functioti of jo and v, the form of

which may be taken for the present to be known as little as.

that of X and 3^, but to which we will give a signification,

which will appear immediately from what follows. Multiply-

ing and dividing the right-hand side of the equation by B,

we have

dQ =^/S&2> + 1^^1)1(3).

We may now choose S, so that the expression in brackets is

a perfect differential ; in other words, so that -„ may be the

integrating factor, or B the integrating divisor, of the expres-

sion within brackets of equation (2)."

From this it follows that in the following equation derived

from (3),

the whole right-hand side is a perfect differential, and there-

fore for a cyclical process we must have

(4),

/f-o (*

In this way Zeuner arrives at an equation similar to equation

(7) of Chapter IV., viz.

{dq_

f= 0.

T

The resemblance, however, is merely external. The essence

of this latter equation consists in this, that t is a function of

temperature only, and further a function which is inde-

pendent of the nature of the body, and is therefore the same

for all bodies. Zeuner's quantity 8, on the contrary, is afunction of both the variables, f and v, on which the bodies'

condition depends ; and further, since the functions X and Y,

in equation (2), are different for different bodies, it must be

true of S also that it may be different for different bodies.

So long as this holds with regard to B, equation (5) has

done nothing for the proof of the second main principle




since it is self-evident that there must in general be an

integrating factor, which may be denoted by -q, and by

which the expression within brackets in equation (2) may

be converted into a complete differential. Accordingly, in

Zeuner's proof, as he himself concludes, everything depends

on the fact that iS* is a function of temperature only, and a

function which is the same for all bodies, so that it may be

taken as the true measure of the temperature.

For this purpose he supposes a body to undergo different

variations, which are such that the body takes in heat whilst

8 has one constant value, and gives out heat whilst 8 hasanother constant value ; and which together make up a cycli-

.cal process, shewing a gain or loss of heat. This procedure

he compares with the lifting or dropping of a weight from

one level to another, and with the corresponding mechanical

work ; and he proceeds (p. 68) : "A further comparison leads

to the interesting result that we may consider the function 8

as a length or a height, and the expression -^ as a weight

in what follows therefore I shall call the above value the

Weight of the Heat." Since a name has here been introduced

for a magnitude containing 8, in which name there is nothing

which relates to the body under consideration, it appears

that an assumption has here been tacitly made, viz. that 8 is

independent of the nature of the body, which is by no means

borne out by the earlier definition.

Zeuner then carries still further the comparison between

the processes relating to gravity and those relating to heat, and

.transfers to the case of heat some of the principles which hold

for gravity ; in so doing he treats 8 as a, height, and -j^ as a

weight, just as before. Then, having finally observed that

the principles thus obtained are true if we take 8 to meanthe temperature itself, he proceeds (p. 74) :" We are there-

fore justified in taking as the basis of our further researches

the hypothesis that 8 is the true measure of temperature."

It appears from this that the only real foundation of the

reasonings, which in bis second edition Zeuner puts forward

as the basis of the second main principle, is the analogy



DISCUSSIONS ON THE THEOBY. 345

between the performance of work by gravity and by heat

and moreover that the point "which has to be proved is in

part tacitly assumed, in part expressly laid down as a mere

hypothesis.

§ 9. RanMne's Treatment of the Subject.

We may now turn to those authors who have considered

that the fundamental principle is not sufficiently trustworthy,

or even that it is incorrect.

Here we must first examine somewhat more closely the

mode of treatment which, as already mentioned, Bankine

considered must be substituted for that of the author.

Rankine, like the author, divides the heat which must be

imparted to the body, in order to raise its temperature, into

two distinct parts. One of these serves to increase the heat

actually existing in the body, and the other is absorbed in

work. For the latter, which comprises the heat absorbed in

the internal and in the external work, Rankine uses an

expression, which in his first section he derives from the

hypothesis that matter consists of vortices. Into this method

of reasoning we need not enter further, since the circum-

stance that it rests on a particular hypothesis as to the

nature of molecules and their mode of motion, makes it

sufficiently clear that it must lead to the consideration ofcom-

plicated questions, and thus leaves much room for doubt as

to its trustworthiness. In the author's treatises he has based

the development of his equations, not on any special viewsas to the molecular constitution of bodies, but only on fixed

and universal principles; and thus, even if the above fact

were the only one which could be alleged against Rankine's

proof, the author would stiU expect his own mode of treating

the subject to be finally established as the most correct.

But yet more uncertain is Rankine's mode of determining

the second part of the heat to be imparted, viz. that which

serves to increase the heat actually existing in the body.

Rankine expresses the increase of the heat within the

body, when its temperature t changes by dt, simply by the

product Kdt, whether the volume of the body changes at the

same time or not. This quantity K, which he calls the real

specific heat, he treats in his proof as a quantity inde-

pendent of the specific voltime^ Any sufficient ground for this



346 ON THE MECflANlCAL THEORY OF HEAT.

procedure will be sought in vain in his paper ; on the contrary,

data are found which stand in direct opposition to it. In the

introduction to his paper he gives, under equation (13), an

expression for the real specific heat K, whichcontains

afactor k, and of which he speaks as follows*: "The co-

efficient Ic (which enters into the value of the specific heat)

being the ratio of the vis viva of the entire motion impressed

on the atomic atmospheres by the action of their nuclei, to

the vis viva of a peculiar kind of motion, may be conjectured

to have a specific value for each kind of substance, depending

in a manner yet unknown on some circumstance in the con-

stitution of its atoms. Although it varies in some cases for

the same substance in the solid, liquid, and gaseous states,

there is no experimental evidence that it varies for the same

substance in the same condition." Hence it appears to be

Rankine's view that the real specific heat of the same sub-

stance may be different in difi:erent states of aggregation

and even for the assumption that it may be taken as in-

variable for the same state of aggregation he produces no

other ground than that there is no experimental proof to the

contrary.

In a later work, A Manual of the Steam Engine and other

Prime Movers, 1859, Rankine speaks yet more distinctly on

this point as follows (p. 307): "A change of real specific

heat, sometinles considerable, often accompanies the change

between any two of those conditions" (i.e. the three con-

ditions of aggregation). How great a difference Rankineconceives to be possible between the real specific heats of one

and the same substance in different states of aggregation, is

shewn by his remark at the same place, that in the case of

water the specific heat as determined by observation, which

he calls the apparent specific heat, is nearly, equal to the real

specific heat. Now Rankine well knew that the observed

specific heat for water is twice as great as that for ice, and

more than twice as great as that for steam. Since then the

real specific heat for ice and steam can never be greater than

the observed, but only smaller, Rankine must assume that

the real specific heat of water exceeds that of ice and steam

by 100 per cent, or more.

If we now ask the question how on this supposition the

* Phil. Mag., Ser. 4, Vol. vn. p. 10. •



DISCUSSIONS ON THE THEORY. 347

increase of heat actually present in a body, whicli occurs

when its temperature t increases by dt, and its volume v bydv, is to be expressed, the answer will be as follows : Whenthe body, during its

change of volume, suffers no change inits state of aggregation, we shall be able to express the

increase of heat, as Rankine has done, by a simple product

of the form Kdt ; but the factor K must have different values

for different states of aggregation. In cases, however, wherethe body during its change of volume also changes its state

of aggregation (e.g. the case we have treated so often, wherewe have a certain quantity of matter partly in the liquid and

partly in the gaseous condition, and where the magnitude of

these two parts changes with the change in volume, either bythe evaporation of part of the liquid, or by the condensation

of part of the vapour), we can then no longer express the

increase of heat connected with a simultaneous change in

temperature and volume by a simple product Kdt ; but mustuse an expression of the form

Kdt + K^dv.

For if the real specific heat of a substance were different in

different states of aggregation, it would be necessary to con-

clude that the quantity of heat existing in it must also

depend on its state of aggregation ; so that equal quantities

of the substance in the solid, liquid, and gaseous condition

would contain different amounts of heat. Accordingly, if

part of the substance change its state of aggregation withoutany change of temperature, there must also be a change in the

quantity of heat contained in the substance as a whole.

Hence it follows that Bankine by his own admission can

only treat the mode in which he expresses the increase of heat,

and the mode in which he uses that expression in his proof, aS

being allowable for the cases in which there are no changes

in the state of aggregation ; and, therefore, his proof holds

for those cases only. For all cases where such changes occur

the principle remains unproved ; and yet these cases are of

special importance, inasmuch as it is chiefly to these that the

principle has hitherto been applied.

In fact we must go further, and say thtii the proof thus

loses all strength even for cases where there is no change in

!the state of aggregation. If Rankine .assumes that the real




specific heat may be different in different states of aggrega-

tion, there seems no ground whatever left for supposing

that it is invariable in the same state of aggregation. It is

known that with solid and liquid bodies changes may occur

in the conditions of cohesion, apart from any change in the

state of aggregation. With gaseous bodies also, in addition

to their great variations in volume, we have the distinction,

that the more or less widely they are removed from their

condensation-point the more or less closely do they follow the

law of Mariotte and Gay-Lussac. How then, if changes in

the state of aggregation may have an influence on the real

specific heat, can we refuse to ascribe a similar, even if

a smaller, influence to changes like the above? Thus the

proposition, that the real specific heat is invariable in the

same state of aggregation, is not only left unproved by

Rankine, but, if we accept his special assumption, becomes in

a high degree improbable.

To this criticism on his proof, which appeared in a

paper of the author's, published in 1863*, Rankine made no

reply; but in a later article on the subject-f* he expressly

maintained the truth of his view, firequently before stated,

that the real specific heat of a body may be different in

different states of aggregation; whereby the force of his

proof is limited to the cases in which no change in the state

of aggregation takes place.

§ 10. Hvriis Objection.

A yet more definite attack upon the author's funda-

mental principle, that heat cannot of itself pass from a

colder to a hotter body, was made by Him in his work, pub-

lished in 1862, Exposition ATudytique et Eocperimentale de la

theorie mecanique de la chaleur, and in two subsequent articles

in Cosmosl. He has there described a particular operation

which gives at first sight an altogether startling result. After

a reply irom the author §, he explained || his attack as having

for its object only to mark an apparent objection to the

principle, whilst in reality he agreed with the author; and

he has expressed himself to the same effect in the second

and third editions of his valuable work.

* Pogg. Ann., Vol. cxx. p. 426.

+ Phil. Mag., Ser. 4, Vol. xxx. p. 410.

J Tol. XXII. pp. 283, 413, § Vol. xxii. p. 560. || VoL xxii. p. 734.




, In spite of this it seems worth while to state here the

attack and the reply, because the conception of the subject

there expressed is one very near the truth, and which might

easily hold under other circiimstances.

Anobjection thus

raised has a real scientific value of its own ; and when it is

put in so clear and precise a light, as Hirn has done in this

case by means of his skilfully-conceived operation, it can

only be advantageous for science : since the fact that the

apparent objection is defined and placed clearly in view will

greatly facilitate the clearing up of the point. In this way

we shall attain the advantage that a difficulty, which other-

wise might probably lead to many misunderstandings, andnecessitate repeated and long discussions, will be disposed of

once and for ever. In^ thus referring once more to this

question, the author is far from wishing to make the objec-.

tion a ground of complaint against Hirn, but rather believes

that this objection has increased the debt which the Mechani-

cal Theory of Heat owes to him on other accounts.

The operation alluded to, on which Hirn has based his

observations, is as follows : Let there be two cylinders A and

B (Fig. 32) of equal area, which are

connected at the bottom by a compara-

tively narrow pipe. In each of these let

there be an air-tight piston; and let

the piston-rods be fitted with teeth

engaging on each side with the teeth

of a spur wheel, so that if one piston

descends the other must rise through

the same distance. The whole space

below the cylinders, including the

connecting pipe, must thus remain

invariable during the motion, because

as the space diminishes in one cylinder

it increases in the other by aa equal

amount.

First, let us suppose the piston in

£ to be at the bottom, and therefore

that in ^ at the top ; and let cylinder

A be filled with a perfect gas of any

given density and of temperature t,.

Now let the piston descend ia A, andFig. 32,




rise in B, so that the gas is gradually driven out of A into

B. The connecting ' pipe through which it must pass is

kept at a constant temperature t^, which is higher than i„, so

that the gas in passing is heated to temperature t^, and at

that -temperature enters cylinder B. The walls of both

cylinders, on the other hand, are non-conducting, so that

within them the gas can neither receive nor give off heat,

but can only receive heat from without as it passes through

the pipe. To fix our ideas let the initial temperature of the

gas be that of freezing, or 0°, and that of the connecting pipe

100°, the pipe being surrounded by the steam of boiling water.

It is easy to see what will be the result of this operation.The first small quantity of gas which passes through the pipe

will be heated from 0° to 100°, and will expand by the cor^

responding amount, i. e. ^^ of its original volume. By this

means the gas which remains in A will be somewhat com-

pressed, and the pressure in both the cylinders somewhat

raised. The next small quantity of gas which passes through

the pipe will expand in the same way, and will thereby com^

press the gas in both cylinders. Similarly each successive

portion of gas will act to compress still further not only the

gas left in A, but also the gas which has previously expanded

in B, so that the latter will continually tend to approach its

initial density. This compression causes a heating of the gas

in both cylinders ; and as all the gas which enters B enters

at a temperature of 100°, the subsequent temperature must

rise above 100°, and this rise must be the greater, the more

the gas within B is subsequently compressed.

Let us now consider the state of things at the end of the

(pperation, when all the gas has passed from A into B. In

the topmost layer, just under the piston, will be the gas

which entered first, and which, as it has suffered the greatest

subsequent compression, will be the hottest. The layers

below will be successively less hot down to the lowest, which

will have the same temperature, 100°, which it attained inpassing the pipe. For our present purpose there is no need

to know the temperature of each separate layer, but only the

mean temperature of the whole, which is equal to the temr

perature that would exist if the temperatures in the different

layers were equalized by a mixing up of the gas. This meantemperjiture will be about 120°.



DISCUSSIONS ON THE THKOEY. 351

In a later article published in Cosmos, Him lias com-

pleted this operation, by supposing that the gas inB is finally

brought into contact with mercury at 0°, and thereby cooled

back again to 0°; that it is then driven back from B to Aunder the same conditions as from A to B, and is therefore

heated in the same manner ; that it is then again cooled by

mercuTy, again driven fromA to B, and so on. Thus we have

a periodical operation, in which the gas is continually brought

back to its original condition, and all the heat given off by

the source of heat passes over to the mercury employed for

cooling. Here we will not enter into this extension of the.

process, but confine ourselves to the first simple operation,in which the gas is heated from 0" to a mean temperature of

120°; for this operation comprises the essence of Hirn's ob-

jection.

In this operation it is clear that no heat is gained or lost

for the pressure in both cylinders is always equal, and there-

fore both pistons are always pressed upwards with equal

force. These forces are communicated to the wheel which

gears with the piston-rods ; and thus, neglecting friction, anindefinitely small force will be sufficient to turn the wheel in

one or the other direction, and thereby move one piston up

and the other down. The excess of heat in the gas cannot

therefore be created by external work.

The process, as is easily seen, is as follows. Whilst a quan-

tity of gas, which is a very small fraction of the whole, is heated

and expandedin passing through the pipe, it must take in

from the source sufficient heat to heat it at constant pressure.

Of this, one part goes to ' increase the heat actually existing

in the gas, and another part to do the work of expansion.

But since the expansion of the gas within the pipe is followed

by a compression of the gas within the cylinders, the same

quantity of heat will be generated in the one place as is

absorbed in the other. Thus that part of the heat derived

from the source, which is turned into work within the pipe,

appears again as heat within the cylinders; and serves to

heat the gas left in A above 0°, its initial temperature, and

the gas which has passed into B above 100°, the temperature

at which it entered ; in other words to produce the rise in

temperature already mentioned. Accordingly, without con-

sidering the intermediate process, we may say that all the



352 ON THE MECHANICAti THEORY OF HEAT.

heat, which the gas contains at the end of the operation

more than at the beginning, comes from the source of heat

attached to the connecting-pipe. Hence we have the sin-

gular result, that by means of a body whose temperature is

100°, i.e. the steam surrounding the pipe, the gas within the

cylinders is heated above 100°, or, looking only to the mean

temperature, to 120°. Here then, is a contradiction of the

fundamental principle that heat cannot of itself pass from a

colder to a hotter body, inasmuch as the heat imparted by

the steam to the gas has passed from a body at 100° to a

body at 120°.

One circumstance however has been forgotten.If the

gas had had an initial temperature of 100° or more, and

had then been raised to a still higher temperature by steam,

whose temperature was only 100°, this would no doubt be a

contradiction of the fundamental principle. But this is not

the real state of things. In order that the gas may be above

100° at the end of the operation, it must necessarily be below

100° at the beginning. In our example, in which the final

temperature is 120°, the initial is 0°, The heat, which the

steam has imparted to the gas, has therefore served in part

to heat it from 0° to 100°, and in part to raise it from 100° to

120°. But the fundamental principle refers only to the

temperatures of the bodies between which heat passes, as

they are at the exact moment of the passage, and not as they

are at any subsequent time. Accordingly we must conceive

the passage of heat in this operation to take place as follows.

The one part of the heat given off by the steam has passed

into the gas, whilst its temperature was still below 100°, and

has therefore passed into a colder body ; and only the other

part, which has served to heat the gas beyond 100°, has passed

into a hotter body. If then we compare this with the funda-

mental principle, which says that, when heat passes from a

colder to a hotter body, without any transformation of work

into heat or any change in molecular arrangement, then ofnecessity there must be in the same operation a passage of

heat from a hotter to a colder body, we easily see that

there is complete agreement between them. The peculiarity

in Him's operation is only this, that there are not two differ-

ent bodies concerned, of which one is colder and the other

hotter than the source of heat ; but one and the same body^



DISCUSSIOKS ON THE THEORY. 353

the gas, takes in one part of the operation the place of thecolder, and in another part that of the hotter body. This in-

volves no departure from the principle, but is only one specialcase out of the many which may occur. Dupre has raised

similar objections against the fundamental principle ; but asthere is nothing in them essentially new, they will not here beentered upon.

§ 11. Wands Objections.

Some years later Th. Wand treated of the same principle

in a paper entitled "Kritische Darstellung des zweiten Satzes

der Mechanische Warmetheorie*." He gives his conclusionsin the three propositions following: (1) "The second prin-

ciple of the mechanical theory of heat, i.e. the impossibility

of a passage of heat to a higher temperature without a conver-

sion into work or a corresponding passage of heat to a lower

temperature, is false." (2) "The deductions from this prin-

ciple are only approximate empirical truths, which hold only

so far as they are established by experiment." (3) "For

technical calculations the principle may be taken as correct,

since experiments on the substances used for the generation

of work and of cold shew a very close agreement with it."

The placing of such propositions side by side seem.i in

itself a doubtful measure. If a principle has been found to

agree with fact in so many cases, as to compel us to say that

for technical calculations it may be taken as correct, it seems

dangerous to conclude nevertheless that it is false, inthe

face of the probable supposition that the apparent objections

which yet remain would be cleared up by closer examination.

The following appear to be the chief grounds on which

Wand bases his rejection of the principle; excluding those

which relate to internal work and electrical phenomena, be-

cause these subjects are not here treated of.

" If we suppose," he says on p. 314, " that in the bring-

ing of a certain quantity of heat from a lower to a highertemperature a certain quantity. of work must of necessity be

destroyed, it follows that, if the same quantity of heat falls

from a higher to a lower temperature, the same quantity of

work must re-appear. Now suppose that this fall takes

• Karl's Repertorium der Bxjtenmental-Phynk, Yol. iv. pp. 281, 369.

c. 23



354 ON THE MECHANICAL THEOET OF EEAT.

place by simple conduction or by a non-reversible cyclical

process. Then the above is not true, because the falling of

heat by conduction goes on without any other change what-

ever. Therefore for equalizations of temperature by simple

conduction there is nothing equivalent to the second prin-

ciple ; and this from the logical point of view is one of the

weakest points in that principle, and leads to much subse-

quent inconvenience." The circumstance here mentioned,

that compensation is required only in the passage of heat to

a higher temperature, and not to a lower, has been frequently

stated above ; and in Chapter X. is expressed in the general

form, that negative transformations cannot take place with-

out positive, but that positive transformations can take place

without negative. From this circumstance the second main

principle becomes doubtless less simple in form than the

first, but it would be hard to shew that it is logically im-

perfect.

The inconveniences mentioned by Wand iri the above

paragraph he arrives at by the following considerations. He

supposes a simple cyclical process to be carried out, duringwhich the two bodies between which the heat passes, and

which he calls the heating and cooling body, have tempera-

tures which are close to 0°, and difier from each other by

an indefinitely small quantity, which he calls dt. For this

symbol, which will appear with another signification in the

analysis which follows, we will substitute h, and will call the

temperatures of the two bodies, reckoned from freezing-point,

and S respectively. Further, Wand supposes the cyclical

process to be so arranged, that one unit of heat passes over

from the hotter to the colder body, and therefore that the

quantity of heat ^^ is transformed into work. He then pro-

ceeds :" The process being ended, I will heat the whole ap-

paratus, comprising both the heating and the cooling body,

by 100°. The difference of temperature between the twobodies will remain unaltered. If we now wish to destroy the

heat thus obtained by means of the reverse cyclical process,

we must take from the colder body the heatf^f. The colder

body thus loses the heat ^^, and giv.es it up to the hotter

body ; and if by the reverse process all is cooled back again

to the initial temperature 0", the initial condition of things



DISCUSSIONS ON THE THEOKT. 355

is restored. There is no work performed or consumed, and

yet a passage of heat has taken place from the second body,

which has remained the colder, to the first which has remained

the hotter throughout the process. This however is no refu-

tation of the second principle. For to obtain this result there

must be a continuous succession of alternate heatings and

coolings of the apparatus, i.e. heat must pass from a hotter

to a colder body; but this passage takes place by conduction,

for which there is no equivalent. Hence it follows from the

process here described, that with regard to the distribution

of the heat it is by no means the same thing, whether wedo nothing at all, or carry out a

compoundcyclical process as

here described."

We have here the case of two opposite cyclical processes

carried out at different temperatures, in which the work

done and the work consumed cancel each other, but more

heat passes from the hotter to the colder body than vice

vers4 ; and "Wand holds that the passage of the surplus heat

from the colder to the hotter body has taken place without

compensation. He has however neglected certain differences

of temperature, which occur in this somewhat complicated

operation. For after the first process, in which the hotter

body has given off and the colder body taken in heat, he

heats the whole apparatus and the two bodies by 100°; and

he cools them by 100° after the second process, in which

the colder body has given off heat and the hotter taken

it in. But in giving off and taking in heat the two bodies

alter their temperature somewhat, and the reservoirs of heat,

which perform the heating and cooling, do not therefore

take back the heat during the cooling at the same tempera-

ture as they gave it out during the heating. Hence arise

passages of heat of which Wand has taken no account.

These differences of temperature are of course very small,

since the two bodies must be assumed so large, that the varia-

tions of temperature produced in them by the cyclical

process may remain small compared to the difference of

the original temperatures. But then the quantities of heat,

which the bodies take from and give back to the reservoirs

during their heating and cooling, are also very large; and

since to determine the heat which has passed we must

multiply the differences of temperature by the actual

23—3



356 OK THE MECHANICAL THEORY OF HEAT.

quantities of heat, we arrive at magnitudes wliicli are quite

large enough to compensate for the surplus heat which has

passed between the bodies.

To prove this point we will make the calculation itself.

First, with regard to the actual surplus heat which has passed

between the bodies, since the temperatures are and B, and

the quantity of heat equals -î^ , this has the equivalence

value2^g (273 + 8

"273)

°^' ^^S^^^^^^S terms of a higher

order thanthe first with regard to S, —

^=^S.

Wehave now

to determine the equivalence value of the passages of heat,

which take place during the heating and coohng of the body

by 100°. By Chapter IV., § 5, we must divide the element

of heat taken in by one of the two bodies from a reservoir of

heat (reckoning heat given off as negative heat taken in)

by the absolute temperature which the body has at the

moment, and we must then form the negative integral for

the heating and cooling.

Let M be the mass of each body, and C its specific heat,

which we suppose constant ; then the quantity of heat,

which it takes in during a rise of temperature dt, equals

MCdt, and this we shall take as expressing the element of

heat. If for convenience we put

^~MG (6).

the element will be expressed by - dt. We must considere

MG as very large, and therefore e as very small, so much

so as to be small even in comparison with the small difiference

of temperature h.

If we now take the first cyclical process, the colder body

has, to begin with, the temperature 0, and the hotter the

temperature S. During the process the former takes in thes

quantity of heat 1, and the latter loses the quantity 1 + ^=^

These must be divided by MC, or multiplied by e, to obtain




the changes of temperature which they produce in the

bodies ; thus at the end of the process 'wie colder body has

the temperature e, and the hotter the temperature

From these temperatures both bodies are now to be heated

by 100°.

The negative integral relating to the heating of "the colder

body is

/lOO+e 1

^"~i. 273+T

If we put T = t — e, then

/lOO

A

.100 1

h 273 H273 + T + e*

Neglecting higher terms in the expansion of e, we have

1 1 6

273 + T + e~273+T (273+t)'"

whence ^ =

--J^273T-r+Jo

(273T^^

^'^-

The negative integral relating to the heating of the

hotter body is

rioo+!-(i+4)' 1

f"

dte

.(,,i-). 273 + r

Whence we obtain in the same way as before

„_ 1/•""+» dr / B Ui°«+ ° cZt .

^7L 273T^'"l^"^273JJs (273 + Tf

^''^-

During the second cyclical process the colder body gives

373off a quantity of heat = gs^ , and the hotTter body receives a



358 ON THE MECHANICAL THEOBY OP HEAT.

quantity ^r^^ + ^f^ . The temperatures of the two bodies,

after the second cyclical process, become respectively

100 + e-||. = 100-|«-«e,

100+S-(l+4)e+(g|+4)e = 100 + S +100

273^'

From these temperaturesboth bodies are cooled down by

100°. The negative integral relating to this cooling is for

the colder body

f 1""1 Ann lOI 1

— ~\ 100^ = I 100

^

'"o-are" 273+^ •'"are' 273+i73+

whence as before we have

^~eJo 273 + T + 273J0 (273 + t)^^''^•

For the cooling of the hotter body we have similarly

1 |-ioo+8 dr _ 100 /•"°+» dr ,

^~ejs 273 +T 273 is (273 + t7••••^^"^•

By adding together A, B, G, D, we obtain the equivalence

value of all the transferences of heat during the heating and

cooling. In this addition the integrals which have the factor

- cancel each other, and two of the others may be combined

together. Whence we have

A+JB + G + I)--

373 /•I"" dr

''27BJo (273 + T)''

V273^273/'J8 (273 + t)'^^^>




Performing the iategration, the right-hand side becomes

173 /_ _1_ 1_\ _ /373 _S_\ /

273 V 373

"*"

27ii) V273"^

273/ [ 373 + S

"^

273 + S.

If we expand the second product in terms of S to the

first order, most of the resulting- terms cancel each other, and

the expression for the equivalence value of the transferences

of heat during the heating and cooling becomes finally

100s

(273)'

*•

This expression fulfils the condition of being equal and

opposite to the equivalence value of the surplus heat

actually transferred between the two bodies. This transfer

therefore is not uncompensated, but fully compensated as the

second principle- requires. We thus see that the operation

suggested by Wand does not give the smallest, ground for

objectingto the

principle.

Another objection is drawn by Wand from the follow-

ing considerations. He proposes the question whether the

principle can be derived by mechanical reasoning from the

ideas which we can form of the nature and action of heat.

With this object he first applies the hypothesis main-

tained by the author and others as to the molecular motion

of gaseous bodies, and finds that this does actually lead

to the principle in question. He then says that it ,is

not sufficient to prove that one particular hypothesis leads

to the principle, bat that all possible mechanical hypo-

theses on the nature of heat must be shewn to lead to it.

Accordingly as a second example he takes another hypo-

thesis, which he conceives specially adapted to represent the

phenomena of expansion, and of the increase of pressure by

heat.

Onthis hypothesis a row of elastic balls, any two of

which are connected by an elastic spring, vibrate in such a

way that all of them are in similar phases. From this

hypothesis he arrives at an equation different from that which

he has chosen as the criterion of the second principle, and

then draws the conclusion, "the second principle cannot

therefore be universally derived from the principles of

mechanics." But upon such a conclusion the question may



S60 ON THE MECHANICAL THEOET OF HEAT.

be asked, -whether in reality the hypothetical motion, which

he supposes, agrees with the actual motion, which we call

heat, in such a way that the same equations must hold for

both. So long as this is not proved the conclusion cannot beconsidered to be made out.

Finally, Wand considers the process which Occurs in

nature, when in the growth of plants, under the influence of

the sun's rays of light and heat, carbonic acid and water are

absorbed, and oxygen liberated ; whilst the organic substances

thus formed, if afterwards burnt or serving as nourishment

to animals, unite themselves again with oxygen to form car-

bonic acid and water, and thereby generate heat. This

transformation of the sun's heat he considers to be in flat

contradiction to the second principle. To the analysis which

he gives many objections might be taken; but the author

considers that a process in which so much is still unknown,

as that of the growth of plants under the influence of the

sun, is altogether unfit to be used as a proof either for or

against the principle in question.

§ 12. Tait's Objections.

Finally the author has to mention certain objections re-

cently raised against his theory by Tait ; objections which have

surprised him equally by their substance and by their form.

In an article which appeared in 1872 on the History of the

Mechanical Theory of Heat*, the author had observed, that

Tait's work, A sketch of Thermodynamics, no doubt owed its

existence chiefly to the wish of claiming the Mechanical

Theory of Heat as far as possible for the English nation ;-â

supposition for which the clearest grounds can be adduced.

Further on in the article he had observed that Tait had

ascribed to Sir William Thomson a formula due to the

author, and had quoted it as given by Thomson in &

paper which contained neither the formula itself nor any-thing equivalent to it. The author expected that Mr Tait,

in answering this article, would specially address himself to

these two points, of which the latter particularly required

clearing up. A reply aj)peared indeed-f,

and one written in

• ôgg. Ann., Vol. cxlv. p. 132.

t Phil. Mag. , Series 4, Vol. xlui.




p. very acrimonious tone ; but to the author's surprise these

two points were nowljere touched upon, a different turn

being given to the whole matter. For whilst in the workpreviously

alluded tothe,

author's researches on the Mechani-cal Theory of Heat, even if in his own view they were not

put in their right relation to those of English writers, were,

yet described at great length and with a general recognition

of their merit, their correctness was here at once assailed, by

the declaration that the fundamental principle, that heat

cannot of itself pass from a colder to a hotter body, is

untrue.

To prove this two phenomena relating to electric currents

are adduced. But in a rejoinder published shortly afterwards *

the author was easily able to prove that these phenomena

in no way contradict the principle, and that one of them is

even so evidently in accordance with it, that it may serve as

an example specially adapted to illustrate and establish it.

As electrical phenomena are not here treated of, this is not

the place to enter further into this subject.

Tait further observed in his reply, that by the author's

introduction of what he namdd Internal Work and Disgrega-

tion, he had done a serious injury to science, offering however

nothing to support this, beyond the brief remark :" In our

present ignorance of the nature of matter such ideas can do

only harm." What Tait has to object to the conception of

internal work, it is difficult to understand. In his first

paper on the Mechanical Theory of Heat the author dividedthe work during the body's change of condition into External

and Internal Work, and shewed that those two quantities of

work differed essentially in their mode of action. Since

that time this distinction has been similarly made by all

writers, so far as he is aware, who have treated of the Mecha-

nical Theory of Heat.

As regards the method (which will be described on a

future occasion) of calculating the combined internal and

external work, and the conception of disgregation introduced,

by the author with this object, purely mechanical investiga-

tions have recently led to an equation, exactly corresponding

with that which the author proposed in the science of heat,

and in which the disgregation makes its appearance. If

* Phil. Mag., Series 4, Yol. xlhi.



36^ ON THE MECHANICAL THEORY OF HEAT.

these investigations cannot yet be considered as complete,

they nevertheless shew, at least in the author's opinion, that

the nature of things requires the introduction of this con-

ception.He therefore leaves the objections of Tait, with as much

confidence as those of Holtzmann, Decher, and others, to the

good judgment of the reader *.

* For a rejoinder by Prof. Tait to these observations, see Sketch ofThermodynamics, 2nd edition, 1877, p. xv.

Finis.



APPENDIX I.

ON THE THERMO-ELASTIC PROPERTIES OP SOLIDS.

Sir William Thomson -was the first who examined the thermo-

elastic properties of elastic solids. Instead of abstracting his

investigation {Quarterii/ Mathematical Jmirnal, 1855) it may be

well to present the subject as an illustration of the method of

treatment by the Adiabatic Function.

Consider any homogeneously-strained elastic solid. To define

the state of the body as to strain six quantities must be specified,

say 11, V, w, X, y, z: these are generally the extensions along three

rectangular axes, and the shearing strains about them, each

relative to a defined standard temperature and a state when the

body is free from stress. The work done by external forces when

the strains change by small variations may always be expressed in

the form

IJJhi+7bv+ ...)x volume of the solid,

because the conditions of strain are homogeneous. U, V are

the stresses in the solid : each is a function oi uv and of the

temperature, and is determined when these are known.. Let 6

denote the temperature (where 6 is to be regarded merely as the

name of a temperature, and the question of how temperatures are

to be measured is not prejudged).

Amongst other conditions under which the strains of the

body may be varied, there are two which we must consider.

First, suppose that the temperature is maintained constant; or

that the change is efiected isothermally. Then 6 is constant.

Secondly, suppose that' the variation is efiected under such con-

ditions that no heat is allowed to pass into or to leave the

body ; or that the change is efiected adidbatically. In the latter'

case 6,u,v, are connected by a relation involving a para-

meter which is always constant when heat does not pass into




or out of the solid : this parameter is called the adiabatic June-,

iion.

We have now fourteen quantities relating to the body, viz.

six elements of strain, six of stress, the quantity 6 which defines

the temperature, and the parameter ^ the constancy of which

imposes the adiabatic condition. Any seven of these may be

chosen as independent variables.

Let the body now undergo Carnot's four operations as fol-

lows :

1°. Let the stresses and strains vary slightly under the sole

condition that the temperature does not change. Let the conse-

quent increase of <^ be 8i^. Heat will be absorbed or given out,and, since the variations are small, the quantity will be propor-

tional to 8(^, say

f{e,u, v,w, .'.....)S<f}.

'2°. Let the stresses and strains further vary adiabatically,

and let S6 be the consequent increase of temperature,

3°. Let the stresses and strains receive any isothermal varia-

tion,such that the parameter ^ returns to its first value. Heat

will be given out or absorbed, equal to

(/+8/)8<^.

4°. Let the body return to its first state.

Here we have a complete and reversible cycle. The quantity

of heat given off SfxS^ is equal to the work done by external

forces. Now Carnot's theorem (or the Second Principle of

Thermodynamics) asserts that the work d6ne, or Sfx 8^, dividedby the heat transferred from the lower to the higher tempera-

ture,/ X 8<^, is equal to a function of 6 only (which function is

the same for all bodies) multiplied by 89. Thus

^=-^(5)8(9,

log (/) = 1^0de + a function of <^

the function of <j> being added because the variation was per-

formed under the condition that ^ was constant. By properly

choosing the parameter <j) this function may be included in 8^,and we have, as the quantity of heat absorbed in the first

operation,

f X bqj.



THERMO-ELASTIC PROPEKTIES OF SOLIDS. 365

The mode of measuring temperature being arbitrary, we shall

find it convenient to define that temperature is so measured that

F (0) = -^-j then we have :

Heat absorbed in first operation = 6S(f> (1);

"Work done by external forces = 86 x Si^ (2).

We must now examine more particularly the variations in the

stresses and strains. Denote the values of U, V, ....... u,v, by

different suffixes for the four operations.

The work done by the external forces in these operations is

respectively

and the sum of these is equal to S(^Sd.

Hence a variety ofimportant relations

maybe obtained.

Let all the strains but one be constant : then we have

M,=



366 ON THE MECHANICAL THEOEY OF HEAT,

__ dll ,. „ dU ,.jj^^^deû, + ^d6

^

„ dU J. „ dU^,

vi__ ,_ae.

Adding these, the total "Work done becomes

du dU du dU\du du du dU\ joji

\-d^M^re-d^)''^^^^'

the differentiations being performed -when u and Zf are expressed

as functions of 6, <j) and the five other strains.

The same is true if five stresses are constant, that is if u and

U are expressed as functions of 6, <;!> and the five other stresses.

But from (2) the "Work done = d6x d<^. Hence it follows

generally (using the well-known theorem as to Jacobians) that

dtf) dO d<j> d9 _^ ,„.

dUd^'d^dU'^ ''

<j) and being expressed as functions of u, U; and either the five

other stresses or the five other strains being constant.

These equations are still true if the independent variables are

partly stresses and partly strains, so long as no two are of the

same name : e.g. if they are vwXYZ.From equation (3) all the thermo-elastio properties of bodies

may be deduced, We have generally

^'=fu'^^fu'^ W'

''^-'P^-fu'^ (^)-





368 ON THE MECHANICAL THEOEY OF HEAT,

These relations are true provided each of the other strains, or

else its corresponding stress, is constant.

Take the last of these for interpretations When 6 is con-

stant we have bj (1),

Heat absorbed in any change (or dq) = 6d<j>.

Hence



APPENDIX II.

ON CAPILLARITY.

The equations obtained in Appendix I. may also be applied to

the eqailibiium of the surface film of a liquid in contact with its

owu vapour. This subject is generally known under the name of

Capillarity, from its having first been studied in connection with

Capillary tubes (Maxwell, On Heat, 1871, p. 263). Thus let the

body considered be the film of fluid at the surface of a fluid, and

let it be so small in volume that its capacity of heat in virtue of

its volume may be neglected. Let T be the surface tension, and

<S^ the surface. Then 2* is a function of 6 only : hence, from an

equation analogous to Equation (3), Appendix L, we obtain

dT dS__

d6"" d^" '

dT d4>__ldg*'''

dO" ^~ edS

(by Equatiion (1*), Appendix I.), where q is heat absorbed.

' dT

If then -j^ is negative,as-

isusually the case, extension of

dd

surface means absorption of heat.

The question of the equilibrium of vapour at a curved sur-

face of liquid has been treated by Sir Wm. Thomson. The

following is mainly taken from his paper {Froc. Royal Society of

-Edinburgh, 1870, Vol. vii., p. 63).

In a closed vessel containing only liquid and its vapour, all

at one temperature, the liq'iiid' restsj with its free surface raised or

depressed in capillary tubes and in the neighbourhood of the solid

boundary, in permanent equilibrium according to the same law of

relation- between curvature and pressure as in vessels open to the

air. The permanence of this equilibrium implies physical equi-

librium between' the liquid and the vapour in contact with it at

all parts of its surface. But the pressure of the vapour at

difierent levels difiers according to hydrostatic law. Hence the

pressure of saturated vapour in contact with ».liquid differs

c. 24




according to the curvature of the bounding surface, being less

when the liquid is concave, and greater when it is convex. And

detached portions of the liquid iu separate vessels, all enclosed in

one containing vessel, cannot remain permanently with their

free surfaces in any other relative positions than those they wouldoccupy if there were hydrostatic communication of pressure

between the portions of liquid in the several vessels. There must

be evaporation from those surfaces which are too high, and con-

densation into the liquid at those surfaces which are too low—process which goes on until hydrostratic equilibrium, as if with

free communication of pressure from vegsel to vessel, is attained.

Thus, for example, if there are two large open vessels of water,

one considerably abpve the other in level, and if the temperatureof the surrounding matter is kept rigorously constant, the liquid

in the higher vessel will gradually evaporate until it is all gone

and condensed into the lower vessel. Or yre may suppose a

capillary tube, with a small quantity of liquid occupying it from its

bottom up to a certain level, to be placed upright in the middle of a

quantity of the same liquid with a wide free surface, and con-

tained in a hermetically sealed exhausted receiver. The vapour

wUl graduallybecome condensed into the liquid in the capillarytube,

untilthe level of the liquid in it is the same as it would be were the

lower end of the tube in hydrostatic communication with the

large mass of liquid. The effect would be that in a very short

time liquid would visibly rise in the capillary tube, and that, pro-

vided care were taken to maintain, the equality of temperature all

over the surface of the hermetically sealed vessel, the liquid in the

capillary tube would soon take very nearly the same level as it

would have were its lower end open ; sinking to this level if the

capillary tube were in the beginning filled too full, or rising to it

if there is not enough of the liquid in it at first to fulfil''the con-

dition of equilibrium.

The following shews precisely the relations between curva-

tures, differences of level, and differences of pressure with which_

we are concerned.

Let jo be the pressure of equilibrium above the curved surface

in the tube, rr' the principal radii of curvature of that surface,

and T the surface tension : then by the principles of Capillarity

the upward drag on the surface, due to the tension, is ^ ( - + -, )

ithin the liquidence the pressure within the liquid, immediately below the

surface is



ON CAPILLARITY. 371

Let V be the equilibrium pressure of th.e vapour at the plane sur-

face of the liquid, p the density of the liquid, tr the density of the

vapour, h the height at which the liquid stands in the tube above

the plane surface : then clearly

-h^^^y +h (1).

But also

IT=p + ha- (2).

Therefore

TT (p- 0-) =^ (p - <r) + r ^-

+-,)

0-,

which gives the relation between the pressure of equilibrium' above

the curved surface and that above the plane surface.

In strictness the value of a- to be uSed' ought to be the mean

density of a vertical column of vapour extending through the

given height. But in all cases in which we can practically

apply the formulae, according to our present knowledge of the pro-

perties of matter, the. difference of densities in this colunm. is very

smaiU, and njiay be' neglected. Hence, if .S denote the height

above the plaij.e surfaee of the liquid of an imaginary homogeneous

fluid whic|i, if of the same density as the vapour at that plane,

would prgduce by its weight the actual pressure jr, we have

j^t^t by Equations (1) and (2)

jiênce by Equation (3)

/IT — ah = Tr(l — -jz.)

For vapour at ordinary atmospheric temperatures, H is about

1,300,000 centimetres. Hence in the capillary tube which would

keep water up to a height ofJ.3 metres above the plane level, the

curved surface of the water is jn equilibrium with the vapour in

contact with it, when the pressure of the vapour is less by about

j^^"' of its own amount than the pressure of vapour in equilibrium

at a plane surfaee pf water ai the same temperature.

24-2



APPENDIX III.

ON THE CONTINUITY OF THE LIQUID ANDGASEOUS STATES OF MATTER.

The chapters on Fusion and Vaporization wDl be rendered more

complete by a brief account of Dr Andrews' important researches

on the continuity between the liquid and gaseous states, of matter.

This account is mainly taken, by permission of Messrs Longmans,

from the late Prof. Maxwell's work on The Theory of Heat, 1877,

Ch. VI.

In Fig. 1 the full lines represent certain Isothermal lines

for Carbonic Acid Gas (CO^), i. e. curves produced by making it

vary in pressure and volume, while the temperature is kept con-

stant at the value written (in degrees Cent.) along each line. Theordinates represent pressures in atmospheres, and the abscissae

represent volumes. The variation is supposed to be produced by

diminishing the volume, e.g. the gas may be supposed to be

contained in a cylinder, the piston of which is gradually lowered.

Take one of these isothermals, e.g. that at 21°'5. Beginning

from the right hand, we have first a curved line AB. During

this part of the compression, the substance is entirely in the state

of a gas. As the volume diminishes the pressure increases, andthe result is a regular curve, which if the gas were perfect, or pv

constant, would be an equilateral hyperbola. At the point B,

corresponding to about 60 atm., the gas commences to liquefy.

From this point the decrease of volume produces no increase of

pressure, but simply liquefies more and more of the gas. Thecurve becomes therefore a horizontal straight line BC. At the

point G the whole of the gas is liquefied. From this point any

further reduction of -^lume is resisted by the elastic force of theliquid, which is very great; and the isothermal line therefore rises

in the almost vertical curve GD. i

Now the length of the horizontal part BG, during which the

gas is partly in the liquid, partly in the gaseous state, is not the

same for different temperatures. For 13°'l its length is seen to be

EF, which is much greater than BG ; while for temperatures above



CONTINUITY OF THE LIQUID AND GASEOUS STATES. 373

21°-5 it is found to be shorter than BO. The dotted line DBEGFis the boundary of all these horizontal lines, up to a certain point G^

Kg. 1.

corresponding to 30'''92, at which their length vanishes. In other-

words, at temperatures above 30''"92 the " gas line " AB, and the



37^! ON THE MECHANICAL THEORY OF HEAT.

"liquid line" CD meet each other, without any intervening,

portion.

In the case of steam, for which such isothermal lines were first

drawn, similar phenomena occur; and when it was seen that the

gas line and the Hquid Une thus continually approached each other

as the temperature was raised, the question naturally arose, Do

they ever meet? If they do meet, then at that temperature the

substance cannot exist partly as a liquid and partly as a vapour,

but must be entirely converted, at the corresponding pressure,

from the state of liquid to that of vapour 3 or else, ?iiice in this

case the vapour and liquid have lihe same density, it may be

suspected that the distinction between liquid and vapour has here

lost its meaning.The answer to this question, has lêen to a great extent sup-

plied by a series of very interesting researches.

In 1822 M. Cagniard de la Tour ' observed the effect of a high

temperature upon liquids enclosed in glass tubes of a capacity not

much greater than that of the liquid itself. He found that when

the temperature was raised to a certain point, the. substance,

which till then was partly Ijquid and partly gaseous, suddenly

became uniform in appearance throughput, without any visiblesurface of separation, or any evidence that the substance in the

tube was partly in one state a:^d partly ii). another.

He concluded that at this temperature the whole became

gaseous. The true conclusion, as Dr Ai?drews has shewn, is,that

the properties of the liquid a^d those pf the vapour continually

approach to similarity, and thafc above * certain temperature, the

properties of the liquid are not separated from those of the vapour

by anyapparent

distinctionbetween^them.

In 1823, the year following the researches of Cagniard de la

Tour, Faraday succeeded in liquefying several bodies hitherto

known only in tê gaseous form, by p-essure alog-e, and in 1826

he êatly extended our knowledge of the effects of temperature

tlflid pressure on gases. He considers that above a certain tempe-

r^jtsire, which, ifl. the language of Dr Andrews, "sre may call the

(jrijtical temperature for the substance, po amount of pressure will

ppodiice the phenomenon which we pÛ condensation, and he

supposes that the temperature of 166° P. below zero is probably

above the critical temperature for oxygen, hydrogen, and nitrogen.

Dr Andrews has examined carbonic acid under varied con-

ditions of temperature and pressure, in order to ascertain the

relations of the liquid and gaseous states, and has arrived at the

conclusion that the gaseous and liquid states are only widely-

' Annalei de Chimie, 2nd Series, Vols. xxi. and xxii.



CONTINUITY OF THE LIQUID AND GASEOUS STATES. 875

separated forms of the same condition of matter, and may be

made to pass one into the other "without any interruption or

breach of continuity '.

The diagram, Kg. 1, for carbonic acid is taken from Dr

Andrews' paper, "with the exception of the dotted line shewingthe region within which the substance can exist as a liquid in

presence of its vapour. The base line of the diagram corresponds,

not to zero pressure, but to a pressure of 47 atmospheres.

The lowest of the isothermal lines is that of 13°'l C. or

55°-6 r:

This line shews that at a pressure of about 47 atmospheres

condensation occurs. The substance is seen to become separated

into two distinct portions, the upper portion being in the state of

vapour or gas, and the lower in the state of liquid. The upper

surface of the liquid can be distinctly seen, and where this surface

is close to the sides of the glass containing the substance it is seen

to be curved, as the surface of water is in small tubes.

As the volume is diminished, more of the substance is liquefied,

till at last the whole is compressed into the liquid form.

Liquid carbonic acid, as was first observed by ThUorier, dilates

as the temperature rises to a greater degree than even a gas, and,as Dr Andrews has shewn, it yields to pressure much more than

any ordinary liquid. Prom Dr Andrews' experiments it also

appears that its compressibility diminishes as the pressure in-

creases. These results are apparent even in the diagram. It is,

therefore, far more compressible than any ordinary liquid, and it

appears from the experiments of Andrews that its compressibility

diminishes as the volume is reduced.

It appears, therefore, that the behaviour of liquid carbonicacid under the action of heat and pressure is vet-y different from

that of ordinary liquids, and ia some respects approaches to that

of a gas.

If we examine the next of the isothermals of the diagram,

that for 21°'5 C. or 70°*7 F., the approximation between the

liquid and the gaseous states is still more apparent. Here con-

densation takes Talsice at about 60 atmospheres of pressure, and

the liquid occupiel nearly a thirdof the volume of the gas. The

exceedingly dense gas is approaching in its properties to the

exceedingly light liquid. Still there is a distinct separation

between the gaseous and liquid states, though we are approaching

the critical temperature. This critical temperature has been

determined by Dr Andrews to be 30»-92 C. or 87''-7 F. At this

temperature, and at a pressure of from 73 to 75 atmospheres,

1 Phil. Trans. 1869, p. 575.




carbonic acid appears to be in the critical condition. No sepa-

ration into liquid and vapour can be detected, but at tbe.same

time very small variations gf pressure or of temperature produce

such great variations of density that flickering movements are

observed in' the tube' "reseflibling in an exaggerated form the

appearances exhibited' during the" mixture of liquids of different

densities, or when columns of heated' air ascend through colder

strata."

Tke isothermal line for' sr'l C. or 88° F. passes above this

critical point. During the whole compression the substance is

never in two distinct conditions in diiSerent parts' of the tube.

When the pressure is less than 73 atmospheres the isothermal

line;- though greatly flatter than that of a perfect gas, resembles

it in- general features. From 73 to 75 atmtisphe^res the' vblume

diminishes very rapidly, but by no means suddenly, and above

this pressure the volume diminishes more gradually than in the

case of a perfect gas, but still more rapidly than in most liquids.

In the isothermals for 32°-5 C. or 90°-5 F. and for SB'-S C. or

9 5°'9 F. we can still observe a slight increase of compressibility

liear the same part of the diagram, but in the isothermal line for

48°"1 C. or 118°'6 F. the curve is concave Upw'ards throughout its

rt'hole course, and differs from the corresponding isothermal line

for a pferfect gas only by being somewhat flatter, shewing thalt for

all ordinary' pressdVes the- volilme is somewhat less than that

assigned- by Boyle's law.

StUlat the temperatui'e of ri8°'6 F. carbonic acid has all the

firoperties of a gas, and the effects of heat and pressure on it differ

from their effects on a perfect gas oiify by quantities reqiiiring

careful experiments to detect them.We have no reason to believe that any^ phenomen'on similar to

condensation would occur, however great" a pressure were applied

to carbonic acid at this temperature.

CAllUllIDGK : miKlED BY C. J. CLAY, M.A., AT IHE UNIVEESITY PKEBS.



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Codl : ITS HISTORY AND ITS USES. By Professors GREfek,

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,

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PHYSICAL SCIENCE.

Gallon—continued.,

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HEREDITARY (iENTOS ; Ajx Inquiry intq its Laws and Con-

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'

duction to the Study "of Physical Phenomena. ByAMôfe

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'

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'• i




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.

PRiMER (i)F BOTANY. With Illustrations., i8mq. is. NewEdition, revised and corrected.

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PHYSICAL SCIENCE.

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the Natural History Sciences:— (6) On the Study of Zoology:—(7) On the Physical Basis

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. . ,



.12 . SCIENTIFIC CATALOGUE.

Lockyer (J.N.)—Worksby J.

Norman Lockyer, F.R.S.—

ELEMENTARY LESSONS IN ASTRONOMY. With nu-

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THE; ORIGIN AND METAMORPHOSES OiF INSECTS.

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PHYSICAL SCIENCE. 13

Macmillan (Rev. Hugh)—B(,«;j««âr,;.'

FIRST FORMS©F VEGETATION. Second Edition, _ corrected

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1,4 SCIENTIFIC CATALOGUE.

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PHYSICAL SCIENCE.15

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PHYSICAL SCIENCE. 17

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'•

Thomson.—Works by Sir Wyville Thomson, K.C.B., F.R.S.

THE DEPTHS OF THE SEA : An Account of the General

Results of the Dredging Cruises of H.M.SS. "Porcupine" and" Lightning " during the Summers of 1868-69 and 70, under the

scientific direction of Dr. Carpenter, F.R.S., J. Gwyn Jeffreys,;

F.R.S., and Sir Wyville Thomson, F.R.S. With nearly 100

Illustrations and 8 coloured Maps and Plans. Second Edition.

Royal 8vo. cloth, gilt. '3,u. 6d.

The Athenseum says :" The book is full of interesting matter, and

is written by a master of the art of popular exposition. It is

excellently illustrated, both coloured maps and ivoodcuts possessing

high merit.. Those who have already become interested in dredging

operations will of course make a point of reading this work ; those

who wish to be pleasantly introduced to the subject, an<l rightly

to appreciate the news which artiyes from time to time from the

' Challenger,' should notfail to seekinstructionfrom it."

THE VOYAGE OF THE " CHALLENGER."—THE ATLAN-TIC. A Preliminary account of the Exploring Voyages of H.M.S.

"Challenger," during the year 1S73 and the early part of 1876.

With numerous Illustrations, Coloured Maps & Charts, & Portrait

of the Author, ^ngraved'.byC. H. JEENS. 2 Vols. Medium 8vo. ,42^.

. I%e Times says—

" It is right that the public should have some

authoritative account of the general results of the expedition, and

that as many of the ascertained data as may be accepted with con-

fidence should speedily find their place in the general body oj

scientific knowledge. No one can ba more competent than the

accomplished scientific chief of the expedition to satisfy thepublic in

this respect The paper, printing, and especiallythe numerous

illustrations, are of- the highest quality; ... We have rarely, if

ever, seen more beautiful specimens ofiiiood engraving than abound

in this work. . . . Sir Wyville Thomson's style is particularly

attractive; he is easy and graceful, but vigorous and exceedingly



PHYSICAL SCIENCE.19

Thomson continued.

hapjyin the choice oflanguage, andtHraughout the work there%retouches which show that science has not banished sentiment .fromhts bosom"

Thudichum and Dupre._A treatise on theORIGIN, NATURE, AND VARIETIES OF WINE.Being a Complete Manual of Viticulture and CEnology. By J. L.W. Thudichum, M.D.., and August Dupr£, Ph.B., Lecturer onChemistry at Westminster Hospital. Medium 8vo. cloth gilt. 25^.

"A treatise almost unique for its usejiilness either to the wine-growerthe vendor, or the consumer of wine. The analyses of wine arethe most complete we have yet Seen, exKibiting at a glance the

constituentprinciples oj nearly all the wines inown in this country.

—Wine Trade Review.

Wâllace (A. R.)—Worksby Alfred Russel Wallace.CONTRIBUTIONS TO THE ' THEORY OF NATURALSELECTION. A Series of Essays. New Edition, with

Corrections and Additions. Crown 8vo. 8s. 6d.

Dr. ffooker, in his address to the British Assofiatian, spokf thus

of the author : "OfMr. Wallace and his many contrOtttions

to philosophical biology it is not easy to speak without enthu-

siasm; for, putting aside their great merits, he, throughout his

writings, with a modesty as rare as I believe it to be uncon-

scious, forgets his awn unquestioned claim to the honour oJ

hoping originated independently of Mr. Darwin, the theories

which he so ably defends." 7%^ Saturday Review says: "Hehas combined an abundarue of fresh and original facts with a

,

liveliness and sagacity of reasoning which are not often displayedso effectively on so small a 'scale." '

THE GEOGRAPHICAL DISTRIBUTION*OF ANIMALS,with a study of the Relations of Living and Extinct Faunas as

Elucidating the Past Changes of the Earth's Surface. 2 vols. 8vo.

with Maps, and numerous Illustrations by Zwecker, 42J. y

The Times says: '^ Altogether it is a wonderful and fascinating

story, whatever objections may be taken to theories founded, upon

it. Mr. IVallace has not attempted to add to its interest by any

adornments of style ; he has given a simple and clear statepient ofintrinsically interesting facts, and. what he considers to be legiti-

mate inductions from them. Naturalists ought to, be grateful to

him for hamng undertaken so toilsome a task. The work, indeed,

is a credit to all concerned—the author, the publishers, the artist—unfortunately now no more—of the attractive illustrations— last

but by no means least, Mr. Stanford's map-designer"




Wallace (A. R.)—continued.

TROPICAL NATURE: with other Essays^ 8vo. I2s.

" Nffwhere amid the niany descriptions of the irdpUs that have been

given is to be found a summary of the past history and actual

phenomena of the tropics which gives thai which is distinctive of

! ; tê .phases \of nature in them more clearly, shortly, and impres-

sively."—Satui'day Review.

Warington,—THE WEEIC OF CREATION; OR, THECOSMOGONY OF GENESIS CONSIDERED IN 'ITS

REL4.TION TO MODERN SCIENCE. By Georg:? Wae-INGTON, Author of " The Historic Character of the Pentateuch

,Vindicate^." Crown 8vo. ,4^. dd.

Wilson.—RELIG16 CHEMICI. By tiie late George Wilson,

M.D., F.R.S.E., Regius Professor of Technology in the University

of Edinburgh. With a Vignette beautifully engraved aftei; a

design by Sir Noel fATON. Crown 8vo. Zs.dd.

"A , more fascinating vpmme," the Spectator says, " has ,seldom

fallen into our hands.", ,

Wilson (Daniel.)—CALIBAN : a Critique on Shakespeare's

"Tempest" and "Midsummer Night's Dream." By DanielWilson,, LL.D., Professor of History and English Literature in

University College, Toronto. 8vo, los. 6d.

'' -The whole volutne is most rich in the eloquence of thought andimagination as well as of words. It is a choice contribution at

once to science, theology, religion, and literature.''''—British

Quarterly Revifew.

Wright.—METALS AND THEIR CHIEF INDUSTRIALAPPLICATIONS. By C. Alder Wright, D.Sc, &c., Lec-

' turei- on Chemistry in St. Mary's Hospital School, Extra fcap.

8vo. y. dd.

-^ Wuftz.—A HISTORY OF CHEMICAL THEORY, from the

Age of Lavoisier down to the present time. By Ad. Wurtz.' Translated by Henry Watts, F.R. Si Crown 8vo.

.6j..,

" The discourse, as a resume of chemical theory and research, unites

singular luminousness and grasp.

A fewjudiciousnotes

are added' by the translator."—Pall Mall Gazette. " The treatment of the

sttbject is admirable, and the translator has evidently done his duly

most efficiently."—'êstimasiex Review.



MENTAL AND MORAL PHILOSOPHY, ETC. 21

WORKS ON MENTAL AND MORALPHILOSOPHY, AND ALLIED SUBJECTS.

Aristotle.—AN INTRODUCTION TO ARISTOTLE'SRHETORIC. With Analysis, Notes, and Appendices. ' By E.

M. Cope, Trinity College, Cambridge. Svo. 14J.

ARISTOTLE ON FALLACIES; OR, THE SOPHISTICIELENCHI. With a Translation and Notes by Edward Poste,

M.A., Fellow of Oriel College, Oxford. 8vo. 8j. 6(f.

Birks.-—Works by the Rev. T. R. BiRKS, Professor of Moral Philo-

sophy, Cambridge :

FIRST PRINCIPLES OF MORAL SCIENCE; or, a Firs

Course of Lectures..delivered in the University of Canibpdge.

Crown Svo. Zs. (yd.. , 1 1

1

' i

ITiis work treats of three topies allpreliminary to the direct exposi-

,. 'Hon of Moral Philosophy. These are the Certainty amd. Dignity

of Mgrai Science, its Spiritual Geography, or relation to vther

main subjects of human thought, and its FormaHve Principles, or

some elementary truths on whuh its whole development must

MODERN UTILITARIANISM; or, The Systems of Paley,

Bentham, and Mill, Examined and Compared. Crown Svo. 6s. 6d.

MODERN PHYSICAL FATALISM, AND THE DOCTRINEOF EVOLUTION ; including ait Examination of Hfefbel-t Spen-

cer's First Principles. Crown Svo, 6s. ,.,._^t-J

Boole. — AN INVESTIGATION OF THE LAWS OFTHOUGHT, OlNf WHICH ARE FOUi^DED -THE

MATHEMATICAL THEORIES OF LOGIC AND 'IPRO-

BABILITIES. By George Boole, LL.D,, Professor ,of

' Mathematics in the Queen's University, Ireland, &c. Svo. ,I4r.''

Butler.—LECTURES ON THE HISTORY OF ANCIENTPHILOSOPHY. By W. Archer Butler, l'at6 Professor of

Moral Philosophy in the University of Dublin. Edited from the

Author's MSS., with Notes/ by William Hepworth ;1Piî",

SON, M.A., Master of Trinity College, and Rfigius PirofeSsbr of

Greek in the Unifersity of Cambridge. New and Cheaper Edftidn,

revised by the Editors Svo. 12s. i;.-;

Caird—A .critical account of the ?HiLpsgteY

.OF KANT. With an. Historical Introduction. By, E. Caird,

M.A,, Professor QfMoral Philosophy in the,University of.Gjasgow.

Svo. iSs. , .,, ,\



22 SCIENTIFIC CATALOGUE.

Calderwood.—Works by the Rev. Henry Calderwood, M.A.,

LL.D., Professor of Moral Philosophy in the University of Edin-

burgh : -,-,-•PHILOSOPHY OF THE INFINITE: A"^ Treatise on Man's

Knowledge of the Infinite Being, in answer to Sir W. Hamilton

and Dr. Mansel. Cheaper Edition. 8vo. Is. 6d.

"A book of great ability > ... written in a clear style, and maybe easily understood by,, even those who are not versed in such

discussions!'—British Quarterly Review-

A HANDBOOK OF MORAL PHILOSOPHY. New, Edition.

Crown 8vo. 6^. .

.

'

_ .,,.'•'

"It is, we/eel convinced, the best handbook on the subject, intellectually

and morally, and does inHnite credit to its author. —Standard,"A- compact and useful work, going over a great deal of ground

in a manner adapted to suggest and facilitate further study. , . .

His book Tjoill be an assistance to many students outside his awnUniversity ofEdinburgh. —Guardian.

THE RELATIONS OF MIND AND BRAIN. INearly ready.

Fiske.—OUTLINES OF COSMIC PHILOSOPHY, BASEDON THE DOCTRINE OF EVOLUTION, WITH CRITI-CISMS ON THE POSITIVE PHILOSOPHY. By JohnFiSKE, M.A., LL.B., formerly Lecturer on Philosophy at

Harvard University. 2 vols. 8vo. ajj.

" The work constitutes a very-ejfecHve encyclopcedia ofthe evolution-

aryphilosophyi and is well worth the study of all who wish to see

at once the entire scope and purport of the scientific dogmatism of

the day."—Saturday Review.

Herbert.—THE realistic assumptions of modernSCIENCE EXAMINED. By T. M. Herbert, M.A., late

Professor of Philosophy, &c., in the Lancashire Independent

College, Msinchester. 8vo. 14J.

Jardine.—THE elements of THE psychology ofCOGNITION. By Robert Jardine, B.D., D'.Sc, Principal of

the General Assembly's College, Calcutta; and Fellow of the Uni-

versity of Calcutta, Crown Svo. 6j. 6<j?.;

Jevons.—Works by W. Stanley Jevons, LL.D., M.A., F.R.S.,

Professor of Political Economy, University College, London.

THE PRINCIPLES OF SCIENCE. A Treatise on Logic and

Scientific Method. New and Cheaper Edition, ieviseflj CrownSvo. I2S. 6d.

" Ifo one in future can be said to have any true''knowledge of what

has been done in the way of logical and scientific method in

England without \ having carefully studied \I¥ofeisor yevons'

book."—Spectator.



MENTAL AND MORAL PHILOSOPHY, ETC. 23

Jevons—continued.

THE SUBSTITUTION OF SIIiIILARS, the true Principle of

Reasoning. Derived from a Modification of Aristotle's Dictum.Fcap. 8vo. zs. 6d.

ELEMENTARY LESSONS IN LOGIC, DEDUCTIVE AND,

jINDUCTIVE. With Questions, Examples, and Vocabulary of

Logical Terms. New Edition. Fcap. 8vo. 3^. 6d.'

PRIMER OF LOGIC. New Edition. l8mo. is.

MaCCOll.—THE GREEK SCEPTICS, from Pyrrho to Sextus.

An Essay which obtained the Hare Prize in the year 1868. By

Norman Maccoll, B.A., Scholar of Downing College, Cam-

biidge. Crown 8vo. 3^. 6d.

M'Cosh—Works by James M 'Cosh, LL.D., President of Princeton

College, New Jersey, U,S. .

^^ He certainly shows himself skilful in that amplication of logic to

psychology, in that inductive science of the human mind which is

the fine side of English philosophy. His philosophy as a whole is

worthy ofattention."—'&.&s'ê de Deux Mondes.

THE METHOD OF THE DIVINE GOVERNMENT, Physical

and Mcaral. Tenth Edition. 8vo. los. 6d.

"This work is distingtîs'hed from other similar ones by its being

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knowledge of its present condition, and by its entering in a

deeperand moreunfettered manner than itspredecessors upon thedis-

cussion oftheappropricUepsychological, ethical, andtheologicalques-

turns. Theauthor keeps aloofat oncefrom the i priori idealism and

dreaminess of German speculation since Schelling, andfrom the

onesidedness and narrowness of the empiricism and positivism

which have so prevailedin England."—Dr. Ulrici, in "Zeitschrift

fiir Philosophic."

THE INTUITIONS OF THE MIND. A New Edition. 8vo.

cloth. 10s. 6d.. J

" The undertaking to adfust the claims of the sensational and in-

tuitionalphilosophies, and of the i posteriori and h. priori methods,

is accontpUshedinthis work with a great amount of success."—, ; IBKeSitminster Review, fI value it for its large acquaintance

'

'with English Philosophy, which has not led him to neglect the

sreat Go-man works. I admire the moderation and clearness, as

well as. comprehensiveness, of the author's views."—Dr. Dbrner, of

Berlin.

AN EXAMINATION OF MR. J: S. MILL'S PHILOSOPHY:

Being a Defence of Fundamental Truth. Second edition, with

additions. lor. bd.

"Such a work greatly needed to be done, and tJuauihor was the man

to doit. This volume is important, not merely in reference to th




M'Cosh—condnueJ.'"'-i'!''"'

mews of Mr. Mitt, but of the mhiple. school of writers, ^pmt andpres'eni, British and Continental, he so ably refiresents."T-:Pxinf;eton

Review.

THE LAWS OF DISCURSIVE THOUGHT : Being a Text-

book of Formal Logic. Crown 8vo. ^s,

'

' The amount of summarized information which it contains is very

great; and it is the only work on the very important subject with

which it deals. Never was such a -work so much needed as in

//«« /ra^Kif ii?y."—London Quarterly Review. '_

CHRISTIANITY 'A]Srb POSITiyiSM :

ASerie? of Lectures to

tlie Times on Natural Theology and Apologetics. Crown 8vo.

']s. (sd.,

, ,

•'. -., ..; ;,.;

THE SCOTTISH PHILOSOPHY FROM HUTCHESON TOHAMILTON, Biographical, Critical, Expository. Royal 8vo. i6j.

Masson.—RECENT BRITISH PHILOSOPHY: A Review

with Criticisms ; including some ' Comments on Mr. Mill's Answer

to Sir William Hamilton. By Davii) Masson, M.A., Professor

of Rhetoric and EnglishLiterature in the University of Edinburgh.

Third Edition, , with an Additional Chapter. Crown 8vo. 6'*.

" We can nowhere point to a work which gives so clear an exposi-

tion of the^ course ofphilosophical speculation in Britain during

thepast century, or which indicates so instructively the mutual in-

fluences ofphilosopihicandscientific thought."—Fortnightly Review.

Maudsley.—Works by H. Maudsley, M.D., Professdr of Medical

Jurisprudence in University College, London.

THE PHYSIOLOGY OF MIND ; being the First Part of a Third

Edition, Revised,, Enlarged, and in great part Rewritten,,of, ,"•ThePhysiology and Pathology of Mind; " Crown 8v6. \os.()d.'

THE PATHOLOGY. OF MIND. llnthe'Press.

BODY AND MIND : an laqtiiry into their Connexion and Mutual

Influence, specially with refeirence to Mental Disorders. AnEnlarged and Revised edition. To which are added, Psychological

Essays. Crown 8vo. ds. dd.

Maurice.—Works by the Rev. Frederick Denison Maurice,

M.A., Professor of Moral Philosophy in the University Of Cam-bridge.;

(For other; 'Vyorks by the same Author, see ThsopogigalCatalqgue.)

':'''J

,

,

'

SOCIAL MORALITY. Twenty-one LectuVes deliveired in the

University of Cambridge. New and Cheaper Editiori. Crowri 8vo.

los.dd.



•MENTAL AND MORAL PHILOSOPHY, ETC. 25

Maurice—continued.

"Whilst

reading it

we are charmed by thefreedomfrom exchisivenessandprejudice, the large charity, the hrfdness ofthought, the eager-ness to recognize and appreciate whateve)' there is of real worthextant in theiiitirU, Which aninifltes it /rom one end to tfie other.

Wegain new thoughtsand new ways ofviewing things, even more,

ferSaps, from being brought for a time under the influence of so

noble and spiritual a mind."—Athenseum.

THE CONSCIENCE : Lectures on Casuistry, delivered in the Uni-versity of Cambridge. New and Cheaper Edition. Crown 8vo. 5j.

T'yJ* Saturday Review says: "We rise from them with detestationof all that is selfish and mean, and with a living impression that

there is such a thing as [goodness after all.

MORAL AND. METAPHYSICAL PHILOSOPHY. Vol. L

,

Ancient Philosophy from the First to the Thirteenth Centuries;

Vol. II. the' FbiTrteenth Century and the French Revolution, witha glimpse into the Nineteenth Century. New Edition andPreface. 2 Vols. 8vo. 25J.

Morgan.ÂNCIENT SOCIETY : or Researches in the Lines of

Human Progress, from Savagery, through Barbarism to Civilisation.

By Lkwis H. Morgan, Member of the National Academy of

Sciences. 8vo. . i6s. -

Murphy.—THE SCIENTIFIC BASES OF FAITH, By

Joseph John Murphy,. Author of " Habit and Intelligence."

8vo. 14J.

'' The booh is, not without,substantial value ; the writer continues the

wofr^ of the best apologists of the last century, it may be with less

'.. forfe and clearness, iftt still with commendable persuasiveness and

tact; andwith an intelligentfeelingfor the changed conditions oj

theproktem."—Academy.

Paradoxical Philosophy.—a Sequel to "The Unseen Uni-

verse." Crown 8vo. "js. 6d.

Picton.

—THE MYSTERY OF MATTER AND OTHER

ESSAYS. By J. Allanson Picton, Author of " New Theories

;and the Old Faith." Cheaper issue with New Preface. Crown

8vo. 6s.

Contents :— ra? Mystery of Matter— Tlie Philosophy of Igno-

rance—The Antithesis of Faith and Sight—The Essential Nature

of Religion— Christian Pantheism.




Sidgwick.—THE METHODS OF ETHICS, , EyjjHENRYSiDGWiCK, M.A., Prselector in Moral and Political Philosophy in

Trinity College, Cambridge. Second 'Editioiiy revised' 'fliiûghoutwith important additions. 8vD. 14J.

A SUPPLEMENT to the First Edition, containing all the important

additions and alterations in the Second. 8vo. 2S.

" This excellent and very mekome volume. .,. . . J-eaving to meta-

physicians any further discussion thai may be needed respecting the

already over-discussedproblem ofthe origin ofthe moralfaculty, he

\ _ tcikes it.for granted' as readily as the geometrician takes spasce for

granted, or thephysicist the existence of matter^ But htjakes Utile

elsefmr granted, and defining ethics as ' th^sciencf of conduct,^ be

carefully examines,'not the various ethnical, sysfems that have been

propounded by Aristotle and AristotUs followers downwards, but

the principles upon which, sofar as they confine themselves to the

strictprovince ofethics, they are based."—AthenSeum.

Thornton. old-fashioned ethics, and common-sense METAPHYSICS, with some of their Applications.,. ByWilliam Thomas Thornton, Autjiorof "ATreatise on Labour."

8vo. los. 6d.

The present volume aeals with problems which are agitating the

minds of all thoughtjul men. The following are the Contents —/. Ante-Utilitarianism. > //. ffistory's Scientific Tretensioiis. JII.

Damd Hume as a MetaphymeHn. IV. HuxkyismU V. Recent

I'hase of Scientific Atheism. VI. Limits ofDemonstrable Theism.

Thring (E., M.A.)—THOUGHTS ON LIFE-SCIENCE,By :Edward Thring, M.a. (Benjamin Place), Head Master of

Uppingham School. „ New Edition, enlarged and revised. Crown

8vo. "js. dd.

Venn.—THE LOGIC OF CHANCE : An Esây on the Founda-

tions and Province of the Theory of Probability, with especial

reterence to its logical bearings, and its apphcation to ^Mdtal and

Social Science. By John Venn, M.A., Fellow and Lecturer of

Gonville and Caius College, Cambridge. Second Edition, re-

written and greatly enlarged. Crown 8vo. lOr. 6d.

" One of the most thoughtful and philosophical treatises on any sub-

ject connected with logic and evidence which has been prudu^d in

this or any other country for many years."—Mill's Logic, vol. ii.

p. 77. Seventh Edition. , ., ,

,

,' - - ,V}:'''.:A






Elementary Science Class-books—continued.

QUESTIONSON LOGKYER'S ELEMENTARY LESSONSIN ASTRONOMY. For the Use of Schools. By John

Forbes Robertson. i8mo, cloth limp. is. Bd.

Physiology.—LESSONS in elementary physiology.With numerous Hlustrations. By T. H. Huxley, F.R.S., Pro-

fessor of Natural History in the Rpyal School of Mines. NeWjEdition. Fcap. 8vo. 4s. 6ii.

'

QUESTIONS OIJ HUXLEY'S PHYSIOLOGY FORSCHOOLS. By T. Alcock, M.D. iSmo. is. (>d.

Botany—LESSONS IN ELEMENTARY BOTANY. ByD.Oliver, F.R.S., F.L.S., Professor of Botany in University

College, London. With nearly Two flundred Illustrations. New,Edition. Fcap. 8vo. d^. dd.

'

Chemistry—LESSONS IN ELEMENTARY CHEMISTRY,INORGANIC AND ORGANIC. By Henry, E. Roscoe,

F.R.S., Professor of Chemistry in Owens College, Manchester,

With numerous Illustrations and Chromo-Litho ôf • the Solar

Spectrum, and of the Alkalies and Alkaline Earths. New Edition.

Fcap. 8vo. 4J-. 6d. ^'

A SERIES OF CHEMICAL ^PROBLEMS, prepared with

Special Reference to the above, by T. E. Thorpe, Ph.D.,

Professor of Chemistry in the Yorfehire College of Science, Leeds.

Adapted for the preparation of Students for the Government,

Science, and Society;of: Arts Examinations. With a Preface byProfessor RoscoE. Fifth Edition, with Key. i8mo. 3J.

,

Political Economy—POLITICAL ECONOMY FOR BE-GINNERS. By MiLLicENT G. Fawcett. New. Edition.

i8mo. 2s.6d..

. •

I r-

Logic—ELEMENTARY LESSONS IN LOGIC ; Deductive andInductive, with {opious Questions and Examples, and a Vocatulary

of Logical Terms. By W. Stanley Jevons, M.A., Professor of

Political Economy in University College, London. New Edition.

Fcap. 8yo. 3^. 6d.

Physics.-^LESSONS IN ELEMENTARY PHYSICS,,;By

Balfour Stevstart, F.R.S., Professor of NaturaLPhilosophy ih

Owens College, Manchester. With, numerous Illustrations aji^

Chromo-Litho of the Spectra 'of the Sun, Stars,',and Nebula;. Ne'wEdition. Fcap.

8vo. 41.6d.

Pi-actical Chemistry.—fHiE OWEJSfS cbLLEGEjUNIQRCOURSE OF PRACTICAL CHEMISTRY. ' By Franci?

,Jones, Chemical Mastfer in the Grammar School, Manch^ter.With 'Preface by Professor RoscoE, and,. Illustrations. • NewEdition. i8mo. ds. 6d.

'






Manuals for St\xA^n\.s—continued.

Flower (W. H.)_aN INTRODUCTION TO THE OSTE-

OLOGY OF THE MAMMALIA. Being the Substance of the

Course of Lectures delivered at the Royal College of Surgeons of

England in 1870. By Professor W. H. Flower, F.R.S.,

F.R.C.S. With numerous Illustrations. New Edition, enlarged.

Crown 8vo. los. 6d.'

'

Foster and Balfour._THE elements of EMBRY-OLOGY. By Michael :Foster, M.D., F.R.S., and F. M.Balfour, M.A. Part I. crown 8vo. ^s. 6il.

Foster and Langley._A COURSE OF ELEMENTARY

PRACTICAL PHYSIOLOGY. By Michael Foster, M.D.,F.R.S., and J. N. Langley, B.A. New Edition; Crown 8vo. 6s.

Hooker (Dr.)_THE STUDENT'S FLORA OF THE BRITISHISLANDS. By Sir

J. D. Hooker, K.C.S.I., C.B., F.R.S.,

M.D., D.C.L. New Edition, revised. Globe Svo. los. 6d,

,Huxley—PHYSIOGRAPHY. An Introduction to the Study of

Nature. By Professor HuxLEV, F.R.S. With numerousIllustrations, and Coloured Plates. New Edition. Crown Svo.

7^. 6d. ' 3'

Huxley and Martin._ACOURSE OF practical IN-STRUCTION IN ELEMENTARY BIOLOGY. By Professor

Huxley, F.R.S., assisted by H. N. Martin, M.B., D.Sc. NewEdition, revised. Crown Svo. 6s.

Huxley and Parker—ELEMENTARY BIOLOGY. PARTII. By Professor Huxley, F.R.S. , assisted by — Parker.With Illustrations.

, \ln>prepdiâtioH.

Jevons.—THE PRINCIPLES OF SCIENCE. A Treatise onLogic and Scientific Method. By Professor W. Stanley Jevons,LL.D., F.R.S., New and Revised Edition. Crown Svo. X2s.6d.

Oliver (Professor)._FlRST BOOK OF INDIAN BOTANY.By Professor Daniel Oliver, F.R.S., F.L.S., Keeper of theHerbarium and Library of the Royal Gardens, Kew. Withnumerous Illustrations. Extra fcap. Svo. 6s. 6d.

Parker and Bettany._THE MORPHOLOGY OF, THESKULL. By Professor Parker and G. T. Bettany. , Illus-

trated. Crown Svo. los. 6d.•

Tait—AN ELEMENTARY TREATISE,

ON HEAT., By Pr«.fessor Tait, F.R.S.E. Illustrated.,'., În the Press.

Thomson.— ZOOLOGY. By Sir C. Wyville Thomson,F.R.S. Illustrated. \In preparation^

Tyler and Lankester.—ANTHROPOLOGY. By E/B.Tylor, M.A., F.R.S., and Professor E. Ray Lankester, M.A.,F.R.S. Illustrated. [In preparation.

Other volumes of these Manuals will follow.



NATURE SERIES.

-^THE SPECTROSCOPE AND ITS APPLICATIONS.By J. N. LOCKYER, F.R>S. With Illustrations. Second Edition. Crown8vo. • 3J. 6</.

THE ORIGIN AND METAMORPHOSES OF IN-SECTS. By Sir JOHN LUBBOCK, M.E., F.R.S. With Illustrations.

Crown8vo.l3j.6rf. Second EdttioH.

THE TR.ANSIT OF VENUS. By G. Forbes, B.A^,Frofeôr of Natuml Philosopliy in the Andersonian Univfersity, GlasgCrw,

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